Parallel Algorithms for Computational Models of Geophysical Systems

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The black solid lines show the three basins, and the dashed lines, the physiographic characteristics (a-g)." ] ] ] "autores" => array:4 [ 0 => array:2 [ "autoresLista" => "Nancy Pérez-Morga, Thomas Kretzschmar" "autores" => array:2 [ 0 => array:2 [ "nombre" => "Nancy" "apellidos" => "Pérez-Morga" ] 1 => array:2 [ "nombre" => "Thomas" "apellidos" => "Kretzschmar" ] ] ] 1 => array:2 [ "autoresLista" => "Tereza Cavazos" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Tereza" "apellidos" => "Cavazos" ] ] ] 2 => array:2 [ "autoresLista" => "Stephen V. Smith" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Stephen V." "apellidos" => "Smith" ] ] ] 3 => array:2 [ "autoresLista" => "Francisco Munoz-Arriola" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Francisco" "apellidos" => "Munoz-Arriola" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714776?idApp=UINPBA00004N" "url" => "/00167169/0000005200000003/v2_201505081406/S0016716913714776/v2_201505081406/en/main.assets" ] "en" => array:19 [ "idiomaDefecto" => true "titulo" => "Parallel Algorithms for Computational Models of Geophysical Systems" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "293" "paginaFinal" => "309" ] ] "autores" => array:1 [ 0 => array:4 [ "autoresLista" => "Antonio Carrillo-Ledesma, Ismael Herrera, Luis M. de la Cruz" "autores" => array:3 [ 0 => array:2 [ "nombre" => "Antonio" "apellidos" => "Carrillo-Ledesma" ] 1 => array:4 [ "nombre" => "Ismael" "apellidos" => "Herrera" "email" => array:1 [ 0 => "iherrera@geofsica.unam.mx" ] "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] 2 => array:2 [ "nombre" => "Luis M." "apellidos" => "de la Cruz" ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Instituto de Geofísica Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México D.F., México" "identificador" => "aff0005" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "*" "correspondencia" => "Corresponding author: Ismael Herrera" ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 619 "Ancho" => 910 "Tamanyo" => 56252 ] ] "descripcion" => array:1 [ "en" => "The original nodes in the coarse-mesh." ] ] ] "textoCompleto" => "1<a name="p294"></a>IntroductionMathematical models of many systems of interest, including very important continuous systems of Earth Sciences and Engineering, lead to a great variety of partial differential equations (PDEs) whose solution methods are based on the computational processing of large-scale algebraic systems. Furthermore, the incredible expansion experienced by the existing computational hardware and software has made amenable to effective treatment problems of an ever increasing diversity and complexity, posed by scientific and engineering applications [PITAC, 2006].Parallel computing is outstanding among the new computational tools and, in order to effectively use the most advanced computers available today, massively parallel software is required. Domain decomposition methods (DDMs) have been developed precisely for effectively treating PDEs in parallel [<a class="elsevierStyleCrossRef" href="#bib0010">DDM Organization, 2012</a>]. Ideally, the main objective of domain decomposition research is to produce algorithms capable of ‘obtaining the global solution by exclusively solving local problems’, but up-to-now this has only been an aspiration; that is, a strong desire for achieving such a property and so we call it ‘the DDM-paradigm’. In recent times, numerically competitive DDM-algorithms are non-overlapping, preconditioned and necessarily incorporate constraints [<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann, 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat et al., 1991</a>; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat et al., 2000</a>; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat et al., 2001</a>; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel, 1993</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel et al., 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur, 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel et al., 2001</a>; <a class="elsevierStyleCrossRef" href="#bib0140">Mandel et al., 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel et al., 2005</a>; <a class="elsevierStyleCrossRef" href="#bib0115">J. Li et al., 2005</a>; <a class="elsevierStyleCrossRef" href="#bib0165">Toselli et al., 2005</a>], which pose an additional challenge for achieving the DDM-paradigm.Recently a group of four algorithms, referred to as the ‘DVS-algorithms’, which fulfill the DDM-paradigm, was developed [<a class="elsevierStyleCrossRef" href="#bib0050">Herrera et al., 2012</a>; <a class="elsevierStyleCrossRef" href="#bib0050">L.M. de la Cruz et al., 2012</a>; <a class="elsevierStyleCrossRef" href="#bib0050">Herrera and L.M. de la Cruz et al., 2012</a>; <a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Carrillo-Ledesma et al., 2012</a>]. To derive them a new discretization method, which uses a non-overlapping system of nodes (the derived-nodes), was introduced. This discretization procedure can be applied to any boundary-value problem, or system of such equations. In turn, the resulting system of discrete equations can be treated using any available DDM-algorithm. In particular, two of the four DVS-algorithms mentioned above were obtained by application of the well-known and very effective algorithms BDDC and FETI-DP [<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann, 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat et al., 1991</a>; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat et al., 2000</a>; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat et al., 2001</a>; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel et al., 1993</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel et al., 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur, 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel et al., 2001</a>; <a class="elsevierStyleCrossRef" href="#bib0140">Mandel et al., 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel et al., 2005</a>; <a class="elsevierStyleCrossRef" href="#bib0115">J. Li et al., 2005</a>; <a class="elsevierStyleCrossRef" href="#bib0165">Toselli et al., 2005</a>]; these will be referred to as the DVS-BDDC and DVS-FETI-DP algorithms. The other two, which will be referred to as the DVS-PRIMAL and DVS-DUAL algorithms, were obtained by application of two new algorithms that had not been previously reported in the literature [<a class="elsevierStyleCrossRef" href="#bib0060">Herrera et al., 2011</a>; <a class="elsevierStyleCrossRef" href="#bib0070">Herrera et al., 2010</a>; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera et al., 2009</a>; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera et al., 2009</a>; <a class="elsevierStyleCrossRef" href="#bib0080">Herrera, 2008</a>; <a class="elsevierStyleCrossRef" href="#bib0085">Herrera, 2007</a>]. As said before, the four DVS-algorithms constitute a group of preconditioned and constrained algorithms that, for the first time, fulfill the DDM-paradigm [<a class="elsevierStyleCrossRef" href="#bib0045">Herrera et al., 2013</a>; <a class="elsevierStyleCrossRef" href="#bib0050">L.M. de la Cruz et al., 2012</a>].Both, BDDC and FETI-DP, are very well-known [<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann, 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat et al., 1991</a>; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat et al., 2000</a>; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat et al., 2001</a>; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel et al., 1993</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel et al., 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur, 1996</a>; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel et al., 2001</a>]; and both are highly efficient. Recently, it was established that these two methods are closely related and its numerical performance is quite similar [<a class="elsevierStyleCrossRef" href="#bib0140">Mandel et al., 2003</a>; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel et al., 2005</a>]. On the other hand, through numerical experiments, we have established that the numerical performances of each one of the members of DVS-algorithms group (DVS-BDDC, DVS-FETI-DP, DVS-PRIMAL and DVS-DUAL) are very similar too. Furthermore, we have carried out comparisons of the performances of the standard versions of BDDC and FETI-DP with DVS-BDDC and DVS-FETI-DP, and in all such numerical experiments the DVS algorithms have performed significantly better.Each DVS-algorithm possesses the following conspicuous features:<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005">•It fulfills the DDM-paradigm;</li><li class="elsevierStyleListItem" id="lsti0010">•It is applicable to symmetric, non-symmetric and indefinite matrices (i.e., neither positive, nor negative definite); and</li><li class="elsevierStyleListItem" id="lsti0015">•It is preconditioned and constrained, and has update numerical efficiency.</li></ul>Furthermore, the uniformity of the algebraic structure of the matrix-formulas that define each one of them is remarkable.This article is organized as follows. In <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a> the basic definitions for the DVS framework are given; here we define the set of ‘derived-nodes’, internal, interface, primal and dual nodes, the ‘derived-vector-space’, among others. <a class="elsevierStyleCrossRef" href="#sec0015">Section 3</a> is devoted to define the new set of vector spaces that conforms the DVS framework; the Euclidean inner product, is also defined here. In <a class="elsevierStyleCrossRef" href="#sec0020">Section 4</a> the ‘transformed-problem’ on the derived-nodes is explained in detail, and this is our starting point to define the DVS algorithms. <a class="elsevierStyleCrossRef" href="#sec0025">Section 5</a> presents a summary of the four DVS-algorithms: DVS-BDDC, DVS-FETI-DP, DVS-PRIMAL and DVS-DUAL. In <a class="elsevierStyleCrossRef" href="#sec0060">Section 6</a> we give the numerical procedures<a name="p295"></a> we use to fulfilling the DDM-paradigm, and we explain in detail the implementation issues. Finally, in <a class="elsevierStyleCrossRef" href="#sec0085">Section 7</a> we show some numerical results obtained after the application of the DVS-algorithms in the solution of several boundary values problems of interest in Geophysics. We studied examples for a single-equation, for the cases of symmetric, non-symmetric and indefinite problems. We also present results for an elasticity problem, where a system of PDE equations is solved.2DVS Framework: A SummaryThe ‘derived-vector-space framework (DVS-framework)’ is applied to the discrete system of equations that is obtained after the partial differential equation, or system of such equations, has been discretized. The procedure is independent of the method of discretization that is used. Thus, the DVS-framework’s starting point is a system of linear algebraic equations that is referred to as the ‘original problem’:<elsevierMultimedia ident="eq0005"></elsevierMultimedia>However, in the DVS setting one does not work with the set of nodes originally used for discretizing the problem the original-nodes’ (<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>). Instead, one uses an auxiliary set of nodes: the ‘derived-nodes’. Each one of such nodes has the property that it belongs to one and only one subdomain of the coarse mesh.<elsevierMultimedia ident="fig0005"></elsevierMultimedia>Indeed, generally after a coarse-mesh has been introduced, some original-nodes belong to more than one subdomain of the coarse-mesh (<a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>), which is inconvenient for achieving the DDM-paradigm. Therefore, in the DVS-framework, each original-node that belongs to more than one subdomain is divided into as many new nodes – the derived-nodes (<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>) - as subdomains it belongs to. Then, the derived-nodes so obtained are distributed into the coarse-mesh subdomains so that each derived-node is assigned to one and only one subdomain of the coarse-mesh (<a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>). Once this has been done, a convenient notation is to label each derived-node by a pair of natural numbers: the first one indicating the original-node from which it derives and the second one, the subdomain to which it is assigned.<a name="p296"></a><elsevierMultimedia ident="fig0010"></elsevierMultimedia><elsevierMultimedia ident="fig0015"></elsevierMultimedia><elsevierMultimedia ident="fig0020"></elsevierMultimedia>The real-valued functions defined in the set of derived-nodes constitute a vector-space: the ‘derived-vector-space’, W. This space becomes a finite-dimensional Hilbert-space when it is supplied with the inner-product that is usually introduced when dealing with real-valued functions defined in a set of nodes; this is referred to as the Euclidean inner-product.Afterwards, a new problem (referred to as the ‘transformed problem’) is defined in the derived-vector-space, which is equivalent to the original system of discrete equations. Thereafter, all the numerical and computational work is carried out in the DVS-space.Before leaving this Section, we dwell a little further on the meaning of a coarse-mesh. By it, we mean a partition of Ω into a set of non-overlapping subdomains {Ω1,...,ΩE}, such that for each α=1, ..., E, Ωα′, is open and:<elsevierMultimedia ident="eq0010"></elsevierMultimedia>Where Ω¯α stands for the closure of Ωα. The set of ‘subdomain-indices’ will be<elsevierMultimedia ident="eq0015"></elsevierMultimedia>Nˆα, α=1,..., E, will be used for the subset of original-nodes that correspond to nodes pertaining to Ω¯α. As usual, nodes will be classified into ‘internal’ and ‘interface-nodes’: a node is internal if it belongs to only one partition-subdomain closure and it is an interface-node, when it belongs to more than one. For the application of dualprimal methods, interface-nodes are classified into ‘primal’ and ‘dual’ nodes. We define:NˆI⊂Nˆ as the set of internal-nodes;NˆΓ⊂Nˆ as the set of interface-nodes;Nˆπ⊂NˆΓ⊂Nˆ as the set of primal-nodes<a class="elsevierStyleCrossRef" href="#fn0005">1</a>; andNˆΔ⊂Nˆ as the set of dual-nodes.The set of primal-nodes is required to be a subset of NˆΓ and, in principle, could be otherwise chosen arbitrarily. However, the algorithms considered by domain decomposition methods are iterative-algorithms and their rate of convergence depends crucially on the selection of the set Nˆπ. Thus, criteria for selecting Nˆπ have been studied extensively (see [<a class="elsevierStyleCrossRef" href="#bib0165">Toselli et al., 2005</a>], for detailed discussions of this topic). Each one of the following two families of node-subsets is disjoint :NˆI,NˆΓ and NˆI,Nˆπ,NˆΔ. Furthermore, these node subsets fulfill the relations:<elsevierMultimedia ident="eq0020"></elsevierMultimedia>Throughout our developments the original matrixA⌢__ is assumed to be non-singular (i.e., it defines a bijection of Wˆ into itself). The following assumption (‘axiom’) is also adopted in throughout the DVS-framework: “When the indices p∈Nˆα and q∈Nˆβ are internal original-nodes, while α ≠ β, then p∈Nˆα and q∈Nˆβ are unconnected”. We recall that unconnected means:<elsevierMultimedia ident="eq0025"></elsevierMultimedia>3The Derived-Vector Space (DVS)In order to have at hand a sufficiently general framework, we consider functions defined on the set X of derived-nodes whose value at each derived-node is a dD-Vector. The numerical applications that will be discussed in this paper correspond to two possible choices of d: when the application refers to a single partial differential equation (PDE), d=1, and for the problems of elasticity that will be considered, which are governed by a three-equations system, d=3.Independently of the chosen value for d, the set of such functions constitute a vector space, W, referred to as the ‘derived-vector space’. When u_;∈W, we write u(p, α) for the value of u_; at the derived-node (p, α). We observe that, in general, u(p, α) itself is a d-Vector and we adopt the notation u(p, α, i), i=1, ..., d. For the i-th component of u(p, α). When d=1 the index i is irrelevant and, in such a case, will deleted throughout.For every pair of functions, u_;∈W and w_;∈W, the ‘Euclidean inner product’ is defined to be<a name="p297"></a><elsevierMultimedia ident="eq0030"></elsevierMultimedia>Here, u(p,α)⊙w(p,α) stands for the inner-product of the dD-Vectors involved; thus,<elsevierMultimedia ident="eq0035"></elsevierMultimedia>A fundamental property of the derived-vector space W, is that it constitutes a finite dimensional Hilbert-space with respect to the Euclidean innerproduct.Let W′⊂W be a linear subspace and assume M⊂X is a subset of derived-nodes. Then, the notation W’(M) will be used to represent the vector subspace of W’, whose elements vanish at every derived-node that does not belong to M. Furthermore, corresponding to each local subset of derived-nodes, Xα, there is a ‘local subspace of derived-vectors’, Wα, which is defined by<elsevierMultimedia ident="eq0040"></elsevierMultimedia>Clearly, when u_;∈Wα⊂W, u_;(p,β)=0 whenever β ≠ α. We observe that<elsevierMultimedia ident="eq0045"></elsevierMultimedia>A derived-vector u_;∈W is said to be continuous when u_;(p,α) is independent of α. The set of continuous vectors constitute the linear subspace, W12.The orthogonal complement (with respect to the Euclidean inner-product) of W12⊂W is W11⊂W. Then W=W11⊕W12. Two projection-matrices a__:W→W and j__:W→W are here introduced; they are the projection-operators, with respect to the Euclidean inner-product on W12 and W11, respectively. When u_;∈W, one has<elsevierMultimedia ident="eq0050"></elsevierMultimedia>the vectors j__u_; and a__u_; are said to be the ‘jump’ and the ‘average’ of u_;, respectively. Therefore, W11 is the ‘zero-average’ subspace, while W12 is the ‘zero-jump’ subspace.Original-nodes are classified into ‘internal’ and ‘interface-nodes’: a node is internal if it belongs to only one subdomain-closure of the coarse-mesh, and it is an interface-node when it belongs to more than one of such closure-subdomains. Some subspaces, significant for our developments, are listed next:<elsevierMultimedia ident="eq0055"></elsevierMultimedia>At present, numerically competitive algorithms need to incorporate restrictions and to this end, in the DVS-framework, a ‘restricted subspace’ Wr⊂W is selected. In the developments that follow, it is assumed that:<elsevierMultimedia ident="eq0060"></elsevierMultimedia>The matrix a__r will be the projection-operator on Wr. We observe that when u_;∈(WI+WΔ), one has a__ru_;=u_;. We also notice that<elsevierMultimedia ident="eq0065"></elsevierMultimedia>4The Transformed ProblemThe transformed-problem consists in finding u_;∈W such that<elsevierMultimedia ident="eq0070"></elsevierMultimedia>Where:<elsevierMultimedia ident="eq0075"></elsevierMultimedia>and<elsevierMultimedia ident="eq0080"></elsevierMultimedia>together with<elsevierMultimedia ident="eq0085"></elsevierMultimedia>The function m (p, q) is said to be the ‘multiplicity’ of the pair (p, q). The ‘derived-nodes’ are created after a coarse-mesh has been<a name="p298"></a> introduced, by dividing the original-nodes as explained in the Overview (<a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>), and then with each ‘derived-node’ we associate a unique pair of numbers (p, α) such that α∈Eˆ and p∈Nˆα. In what follows, we identify derived-nodes with such pairs.Then, in order to incorporate the constraints, we define<elsevierMultimedia ident="eq0090"></elsevierMultimedia>then, the matrix A__:Wr→Wr defined by<elsevierMultimedia ident="eq0095"></elsevierMultimedia>has the property that<elsevierMultimedia ident="eq0100"></elsevierMultimedia>Hence, <a class="elsevierStyleCrossRef" href="#eq0070">Eq. (4.1)</a> is replaced by<elsevierMultimedia ident="eq0105"></elsevierMultimedia>For matrices and vectors the following notation is adopted:<elsevierMultimedia ident="eq0110"></elsevierMultimedia>where the matrices<elsevierMultimedia ident="eq0115"></elsevierMultimedia>furthermore,<elsevierMultimedia ident="eq0120"></elsevierMultimedia>The matrix A__:W→W will be referred to as the ‘transformed-matrix’. We observe that A__=A__t when π = Ø.In turn, the transformed problem of (4.8) can be reduced, see [<a class="elsevierStyleCrossRef" href="#bib0070">Herrera et al., 2010</a>; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera et al., 2009</a>; <a class="elsevierStyleCrossRef" href="#bib0080">Herrera, 2008</a>; <a class="elsevierStyleCrossRef" href="#bib0085">Herrera, 2007</a>; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat et al., 2000</a>] for details, into the following problem, which is expressed in terms of the values of the solution at dual-nodes, exclusively: “Find u_;Δ∈W (∆) that satisfies<elsevierMultimedia ident="eq0125"></elsevierMultimedia>Here, f_;Δ∈a__W(Δ) and the ‘Schur-complement matrix with constraints’ are defined by<elsevierMultimedia ident="eq0130"></elsevierMultimedia>and<elsevierMultimedia ident="eq0135"></elsevierMultimedia>respectively.5The DVS-AlgorithmsGenerally two kinds of approaches are distinguished: primal –these are direct approaches, which do not resort to Lagrange multipliers- and dual –indirect approaches that use Lagrange multipliers-. However, when DDMs are formulated using a setting as general as that supplied by the DVS-framework, such a distinction is irrelevant. The feature that is conspicuous for different options is the information that the algorithm seeks. Indeed, four algorithms will be obtained by seeking successively for the vectors: u_;Δ,S__−1j__S__u_;Δ, j__S__u_;Δ, and S__u_;Δ. However, in the presentation that follows we stick to the ‘primal vs. dual-algorithms’ classification.5.1Primal FormulationsThe DVS Version of BDDCThis is a primal algorithm which seeks directly for u_;Δ. Pre-multiplying <a class="elsevierStyleCrossRef" href="#eq0125">Eq. (4.12)</a> by a__S__−1, one gets:<elsevierMultimedia ident="eq0140"></elsevierMultimedia>In [<a class="elsevierStyleCrossRef" href="#bib0040">Farhat et al., 2000</a>], it was shown that <a class="elsevierStyleCrossRef" href="#eq0140">Eq. (5.1)</a> is equivalent to <a class="elsevierStyleCrossRef" href="#eq0125">Eq. (4.12)</a>. This equation is the DVS-version of BDDC.The DVS-Primal AlgorithmFor this algorithm, the sought-information is:<a name="p299"></a><elsevierMultimedia ident="eq0145"></elsevierMultimedia>Applying to a__S__<a class="elsevierStyleCrossRef" href="#eq0145">Eq. (5.2)</a> it is seen that a__S__v_;Δ=0. Furthermore,<elsevierMultimedia ident="eq0150"></elsevierMultimedia>Therefore<elsevierMultimedia ident="eq0155"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#eq0155">Eq. (5.4)</a> does not define an iterative algorithm. In order to obtain such an algorithm, we project on j__S__−1WΔ, to obtain:<elsevierMultimedia ident="eq0160"></elsevierMultimedia>This algorithm is referred to as the ‘DVS-primal algorithm’. The solution is given by<elsevierMultimedia ident="eq0165"></elsevierMultimedia>We observe that we could have written u_;Δ=v_;Δ+S__−1f_;Δ instead of <a class="elsevierStyleCrossRef" href="#eq0165">Eq. (5.6)</a>. However, the application of the projection operator a__ is important when ν_;Δ and S__−1f_;Δ are not computed with exact arithmetic, as it is the case when using numerical methods, because when it is applied it replaces v_;Δ+S__−1f_;Δ by the continuous-vector closest (with respect to the Euclidean distance) to it.5.2Dual FormulationsThe DVS Version of FETI-DPFor this algorithm the sought-information is defined to be: λ_;Δ≡−j__S__u_;Δ. This algorithm can be easily derived from the DVS-primal formulation that has just been presented. We observe that v_;Δ=S_;−1λ_;Δ,λ_;Δ=S__ν_;Δ, in view of <a class="elsevierStyleCrossRef" href="#eq0145">Eq. (5.2)</a>, and a__λ_;Δ=0. This permits transforming <a class="elsevierStyleCrossRef" href="#eq0160">Eq. (5.5)</a> into<elsevierMultimedia ident="eq0170"></elsevierMultimedia>Applying S__−1 to the first of these equations, it is obtained:<elsevierMultimedia ident="eq0175"></elsevierMultimedia>As for <a class="elsevierStyleCrossRef" href="#eq0165">Eq. (5.6)</a>, it becomes:<elsevierMultimedia ident="eq0180"></elsevierMultimedia>The DVS-Dual AlgorithmIn this algorithm, the sought-information is: μ_;Δ≡S__u_;Δ. Then, u_;Δ=S__−1μ_;Δ. Replacing this in <a class="elsevierStyleCrossRef" href="#eq0140">Eq. (5.1)</a>, one gets:<elsevierMultimedia ident="eq0185"></elsevierMultimedia>Finally, multiplying by S__ the first of these equalities, it is obtained:<elsevierMultimedia ident="eq0190"></elsevierMultimedia>When μ_;Δ is known, u_;Δ can be recovered by means of<elsevierMultimedia ident="eq0195"></elsevierMultimedia>A comment similar to that made immediately after <a class="elsevierStyleCrossRef" href="#eq0165">Eq. (5.6)</a>, goes here: we have applied the projection matrix a__, in <a class="elsevierStyleCrossRef" href="#eq0195">Eq. (5.12)</a> because we are assuming that exact arithmetic generally will not be used.6Numerical Procedures Fulfilling the DDM-ParadigmSummarizing, the preconditioned DVS-algorithms with constraints are: <a name="p300"></a><elsevierMultimedia ident="eq0200"></elsevierMultimedia><elsevierMultimedia ident="eq0205"></elsevierMultimedia><elsevierMultimedia ident="eq0210"></elsevierMultimedia><elsevierMultimedia ident="eq0215"></elsevierMultimedia>6.1Comment on the DVS Numerical ProceduresThe outstanding uniformity of the formulas given in <a class="elsevierStyleCrossRef" href="#eq0200">Eqs. (6.1)</a> to <a class="elsevierStyleCrossRef" href="#eq0215">(6.4)</a> yields clear advantages for code development, especially when such codes are built using object-oriented programming techniques. Such advantages include:<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0020">I.The construction of very robust codes. This is an advantage of the DVS-algorithms, which stems from the fact the definitions of such algorithms exclusively depend on the discretized system of equations, obtained after discretization of the partial differential equations considered (referred to as the original problem), but which is otherwise independent of the problem that motivated it. In this manner, for example, essentially the same code was applied to treat 2-D and 3-D problems; indeed, only the part defining the geometry had to be changed, and that was a very small part of it;</li><li class="elsevierStyleListItem" id="lsti0025">II.The codes may use different local solvers, which can be direct or iterative solvers;</li><li class="elsevierStyleListItem" id="lsti0030">III.Minimal modifications are required for transforming sequential codes into parallel ones; and</li><li class="elsevierStyleListItem" id="lsti0035">IV.Such formulas also permit developing codes which fulfill the DDM-paradigm; i.e., in which “the solution of the global problem is obtained by exclusively solving local problems”.</li></ul>This last property makes the DVS-algorithms very suitable as a tool to be used in the construction of massively-parallelized software, so much needed for efficiently programming the most powerful parallel computers available at present. In the next Subsection, procedures for constructing codes possessing Property IV are explained with some detail.All the DVS-algorithms of <a class="elsevierStyleCrossRef" href="#eq0200">Eqs. (6.1)</a> to <a class="elsevierStyleCrossRef" href="#eq0215">(6.4)</a> are iterative and can be implemented with recourse to Conjugate Gradient Method (CGM), when the matrix is definite and symmetric, or some other iterative procedure such as GMRES, when that is not the case. At each iteration step, depending on the DVS-algorithm that is applied, one has to compute the action on a derived-vector of one of the following matrices: a__S__−1a__S__, j__S__j__S__−1, S__−1j__S__j__ or S__a__S__−1a__. Such matrices in turn are different permutations of the matrices S__, S__−1, a__ and J__. Thus, to implement any of the preconditioned DVS-algorithms, one only needs to separately develop codes capable of computing the action of each one of the matrices S__, S__−1, a__ or j__ on an arbitrary derived-vector, of W.Therefore, next we present numerical procedures for computing the application of each one of the matrices S__, S__−1, a__ and j__ which fulfill the DDM-paradigm. It will be seen that only a__ requires exchange of information between derived-nodes belonging to different subdomains; actually, between derived-nodes that are descendants of the same original-node (the exchange of information is minimal). As for j__=I__−a__ once the action of a__ has been computed, no further exchange of information is required.6.2Application of S__From <a class="elsevierStyleCrossRef" href="#eq0130">Eq. (4.13)</a>, we recall the definition of the matrix S__≡A__ΔΔ−A__ΔΠA__ΠΠ−1A__ΠΔ. In order to evaluate the action of S__ on any derived-vector, we need to successively evaluate the action of the following matrices A__ΠΔ, A__ΠΠ−1, A__ΔΠ and A__ΔΔ. Nothing special is required except for A__ΠΠ−1. A procedure for evaluating the action of this matrix, which fulfills the DDM-paradigm is explained next.We have<elsevierMultimedia ident="eq0220"></elsevierMultimedia>Let ν_;∈W, be an arbitrary derived-vector, and write<elsevierMultimedia ident="eq0225"></elsevierMultimedia>Then, w_;=w_;I+w_;π∈W is characterized by<elsevierMultimedia ident="eq0230"></elsevierMultimedia>and can obtained iteratively. Here,<a name="p301"></a><elsevierMultimedia ident="eq0235"></elsevierMultimedia>and, with a__π as the projection-matrix into Wr(π),j__π≡I__−a__π.We observe that fulfilling the DDM-paradigm when computing the action of A__II−1 is straightforward because<elsevierMultimedia ident="eq0240"></elsevierMultimedia>is parallelizable. Once v_;π∈Wr(π) has been obtained, to derive ν_;I one can apply:<elsevierMultimedia ident="eq0245"></elsevierMultimedia>this completes the evaluation of S__.6.3Application of S__−1We define<elsevierMultimedia ident="eq0250"></elsevierMultimedia>and observe that<elsevierMultimedia ident="eq0255"></elsevierMultimedia>Therefore, the matrix A__−1 can be written as:<elsevierMultimedia ident="eq0260"></elsevierMultimedia>Furthermore, S__:WΔ→WΔ fulfills<elsevierMultimedia ident="eq0265"></elsevierMultimedia>Another property that is relevant for the following discussion is:<elsevierMultimedia ident="eq0270"></elsevierMultimedia>for any ν_;∈W, let us write<elsevierMultimedia ident="eq0275"></elsevierMultimedia>then, w_;π fulfills<elsevierMultimedia ident="eq0280"></elsevierMultimedia>Here, j__r≡I__−a__r, where the matrix a__r is the projection operator on Wr′ while<elsevierMultimedia ident="eq0285"></elsevierMultimedia>Furthermore, we observe that<elsevierMultimedia ident="eq0290"></elsevierMultimedia>In order to use <a class="elsevierStyleCrossRef" href="#eq0290">Eq. (6.19)</a> as a means of parallelizing the DVS-algorithms, however, the detailed discussion of such procedures will be presented separately [<a class="elsevierStyleCrossRef" href="#bib0045">Herrera et al., 2013</a>; L.M. de la Cruz et al., 2013]. It is necessary that the local matrices, A__∑∑α, be invertible. This is granted when invertible A__ in Wr′ which generally is achieved by taking a sufficiently large number of primal-nodes.<a class="elsevierStyleCrossRef" href="#eq0280">Eq. (6.17)</a> is solved iteratively. Once ν_;π has been obtained, we apply:<elsevierMultimedia ident="eq0295"></elsevierMultimedia>This procedure permits obtaining A__−1w_; in full; however, we only need A__−1ΔΔw_;. We observe that<elsevierMultimedia ident="eq0300"></elsevierMultimedia>The vector A__−1w_;Δ can be obtained by the general procedure presented above. Thus, take w_;≡w_;Δ∈WΔ⊂W and<elsevierMultimedia ident="eq0305"></elsevierMultimedia>Therefore, <a name="p302"></a><elsevierMultimedia ident="eq0310"></elsevierMultimedia>6.4Application of a__ and j__.We use the notation<elsevierMultimedia ident="eq0315"></elsevierMultimedia>then [<a class="elsevierStyleCrossRef" href="#bib0070">Herrera et al., 2010</a>]:<elsevierMultimedia ident="eq0320"></elsevierMultimedia>while j__=I__−a__ therefore,<elsevierMultimedia ident="eq0325"></elsevierMultimedia>Therefore, only the evaluation of a__u_; requires exchange of information between subdomains. In general, such numbers are very small; for example in application to single-equation problem, when an orthogonal grid is used, they are at most: 4, for problems in 2D, and 8 for problems in 3D.As for the right hand-sides of <a class="elsevierStyleCrossRef" href="#eq0135">Eqs. (4.14)</a>, all they can be obtained by successively applying to f_;Δ some of the operators that have already been discussed. Recalling <a class="elsevierStyleCrossRef" href="#eq0135">Eq. (4.14)</a>, we have<elsevierMultimedia ident="eq0330"></elsevierMultimedia>The computation of Rf⌢_; does not present any difficulty and the evaluation of the actions of A__ΠΠ−1 and A__ΔΠ were already analyzed.7Numerical ResultsTaking into account the general description of the DVS-framework given of <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>, it can be seen that each one of the DVS-algorithms is uniquely defined by:<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0040">1.The original-matrix;</li><li class="elsevierStyleListItem" id="lsti0045">2.The partition of the set of original-nodes, which is induced by the coarse-mesh that is applied; and</li><li class="elsevierStyleListItem" id="lsti0050">3.The set of constraints.</li></ul>In turn, the original-matrix is determined by the partial differential equation, or system of such equations, the discretization method chosen and the fine-mesh adopted. As explained in <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>, the partition of the set of original-nodes depends when the fine-mesh has already been defined, on the coarse-mesh (i.e., the domain decomposition) used. The coarse-mesh is constituted by a family of non-overlapping subdomains {Ω1,..., ΩE} of Ω, the domain of definition of the boundary-value problem to be solved. In all the examples that are presented in this article, the constraints are fully determined by the primal-nodes and consist in requiring continuity of the derived-vectors at them.Several codes were developed to treat the examples, which were written in C++ language, using the MPI library for the communications. In the computational implementations, the methods of solution used to treat the original-problems are: CGM, when such a linear system is symmetric and positive-definite and GMRES when the discrete system is non-symmetric or indefinite. Both are applied with a tolerance of 10−6. Each DVS-algorithm was applied to each one of the examples considered, except for that referring to elasticity.The results obtained for Examples 1 to 5 are summarized in <a class="elsevierStyleCrossRef" href="#tbl0005">Tables 1</a> to <a class="elsevierStyleCrossRef" href="#tbl0025">5</a>, respectively. In them, the acronym dof stands for to the number of degrees of freedom of the original problem, but it should be mentioned that the procedures used to treat such examples are such that the nodes that lie on the external boundary do not contribute to the dof. The notation to indicate the meshes that were adopted is as follows: In 2D cases, we use (nxm)x(qxr), where (nxm) refers to the coarsemesh, while (qxr) to the fine-mesh; and similarly, in 3D cases, we use (nxmxp)x(qxrxs), where (nxmxp) define the coarse-mesh and (qxrxs) the fine-mesh. The constrains are imposed on the primal nodes, in all of our experiments the primal nodes were located at vertex in 2D and at edges in 3D of the subdomains, this coinciding with the algorithm “D” in [<a class="elsevierStyleCrossRef" href="#bib0165">Toselli et al., 2005</a>].<elsevierMultimedia ident="tbl0005"></elsevierMultimedia><elsevierMultimedia ident="tbl0025"></elsevierMultimedia>Each Table contains at most ten columns. The first four indicate respectively: 1) the meshes used, 2) the number of subdomains of the coarsemesh, 3) the dof, and 4) the number of primal-nodes used. The figures appearing in columns 5 to 9 correspond to the number of iterations that were required for convergence of each one of the algorithms applied. Columns 9 and 10 were only included in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>. For Example 3, in order to cover a wide range of values of the Peclet-number, the diffusion coefficient in <a class="elsevierStyleCrossRef" href="#eq0345">Eq. (7.3)</a>, ν, was varied and the tenth column in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a> indicates the different values of ν for which the corresponding boundary-value problem was solved. Furthermore, the results obtained when the DVS-algorithms were applied were compared with those obtained in [<a class="elsevierStyleCrossRef" href="#bib0005">Da Conceição et al., 2006</a>] for the same problem, using the standard version of BDDC.<a name="p303"></a><elsevierMultimedia ident="tbl0015"></elsevierMultimedia>7.1Application of the DVS-algorithms to a Single-EquationThe applicability of the DVS-algorithms is wide, as previously said it can be applied to general equation systems. In <a class="elsevierStyleCrossRef" href="#sec0015">Section 3</a>, it was n announced that in this paper we present examples for which d, the number of equation s of th e system, is one and three. In this Subsection the examples for which d=1 will be discussed, leaving for the next Subsection the treatment of static-elasticity models, for which d=3.Four boundary value problems corresponding to a single-equation will be presented. The first two are symmetric and positive definite boundaryvalue problems, whose definition involves the Laplace differential operator. The other two correspond to advection-diffusion transport, and the corresponding boundary-value problems are non-symmetric and indefinite. The discretization methods used in this Subsection are based on central finite differences (CFD), which are directly applicable to the symmetric problems. To apply CFD to the advection-diffusion problems it was necessary to stabilize the advection-diffusion differential-operator and to this end artificial diffusion was incorporated.Despite the simplicity of the examples presented in this Subsection, they are very important because a wide range of geophysical systems give rise to similar problems [<a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Pinder, 2012</a>]. The diversity of physical interpretations of the boundary-value problems here discussed is enormous. All the differential operators involved can be classified as advection-diffusion operators, since Laplace operator is obtained from the general advection-diffusion differential-operator when the transport-velocity vanishes. Transport processes of heat and solutes occur in a great diversity of geophysical systems. However, the physical processes governed by such differential-equations go far beyond transport phenomena.Example 1. Poisson equation in two-dimensions.<elsevierMultimedia ident="eq0335"></elsevierMultimedia>We can see from <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>, that the four algorithms perform very well as the number of subdomains and the degrees of freedom (dof) are increased. In this example, the DVS-DUAL algorithm presents the best performance, requiring only 11 iterations from 12x12 until 30x30 subdomains, and the same number of dof. All other algorithms show similar behavior. The numerical solution of this example can be seen in the <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>.<elsevierMultimedia ident="fig0025"></elsevierMultimedia>Example 2. Similar to Example 1, but it is formulated in a 3D domain.<elsevierMultimedia ident="eq0340"></elsevierMultimedia>In <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>, we observe a similar performance of the algorithms as in the two-dimensional case. One more time the DVS-DUAL algorithm presents a little better behavior with respect all others.<elsevierMultimedia ident="tbl0010"></elsevierMultimedia>Example 3. The boundary-value problem treated is:<elsevierMultimedia ident="eq0345"></elsevierMultimedia>This is an advection-diffusion transport problem in 2D, for which the differential operator is not self-adjoint.This example is very interesting because it contains diffusion and advection terms, which are common in several complex geophysics phenomena. In this example, the Péclet number is defined as Pe=b_;L/ν where L is a characteristic length (in this case L = 1). We also define a local Péclet number as Peh=b_;h/ν. Using these definitions, fixing the global partition to h=1/512, and the varying the viscosity from 0.01 to 0.0001, we have that the Péclet number varies from 316 to 316,227, and the local Péclet number varies from 0.617 to 617. In this case the linear system is non-symmetric, therefore we choose the GMRES method with a tolerance of 10−6.In <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a> presents the results that the DVS-algorithms yielded and compares them with those obtained in [<a class="elsevierStyleCrossRef" href="#bib0005">Da Conceição et al., 2006</a>]. We observe that, with fixed coarse and fine meshes, as the viscosity coefficient is reduced, so that the Péclet number increases, generally the iterations required for convergence reduce. Increasing the Péclet number implies that the effect of the advection term enlarges, and the numerical solution generally becomes unstable. However, the performance of the discretization strategy based on CFD combined with stabilization of the numerical-scheme by means of artificial viscosity is resilient to Péclet-number variations. For comparison purposes, the examples presented here were chosen to be the same as those<a name="p304"></a> presented in [<a class="elsevierStyleCrossRef" href="#bib0005">Da Conceição et al., 2006</a>], where the standard BDDC algorithm was applied with the same set of constraints; namely, the same set of subdomains and vertex nodes were chosen to be primal. As can be seen in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>, when the comparison criterion is based on the number of iterations required for convergence, the observed performance of the DVS-algorithms in these examples is slightly better than that of the standard BDDC algorithm. Finally, an illustration of the kind of numerical solution obtained is shown in <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>.<a name="p305"></a><elsevierMultimedia ident="fig0030"></elsevierMultimedia><elsevierMultimedia ident="fig0035"></elsevierMultimedia>The relative-residual decay for a coarse mesh (16×16) and several fine meshes is presented in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>. We consider in these computations b = (1, 3) and ν = 0.00001, in such a way that Pe = 3.16e+5. We observe that the best convergence is obtained when the fine mesh is increased, and the convergence slows when the dof occurring in the subdomains is reduced.<elsevierMultimedia ident="fig0040"></elsevierMultimedia>Example 4. The boundary-value problem treated is:<elsevierMultimedia ident="eq0350"></elsevierMultimedia>This is an advection-diffusion transport problem in 3D, for which the differential operator is not self-adjoint.The diffusion and advection-diffusion differential-operator appears in the equations of the examples presented above. They are very important in natural and industrial phenomena. For example, the flow and transport of solutes in subsurface groundwater, the movement of aerosol and trace gases in the atmosphere, mixing of fluids in processes of crystal growth, among many other important applications [<a class="elsevierStyleCrossRef" href="#bib0160">Tood, 1980</a>; <a class="elsevierStyleCrossRef" href="#bib0150">Pinder et al., 2006</a>; <a class="elsevierStyleCrossRef" href="#bib0095">Herrera et al., 1969</a>; <a class="elsevierStyleCrossRef" href="#bib0100">Herrera et al., 1973</a>; <a class="elsevierStyleCrossRef" href="#bib0105">Herrera et al., 1977</a>; <a class="elsevierStyleCrossRef" href="#bib0110">Herrera G.S. et al., 2005</a>; <a class="elsevierStyleCrossRef" href="#bib0020">L.M. de la Cruz et al., 2006</a>]. In all our examples, we have shown that the DVS-algorithms obtain the numerical solution efficiently on parallel machines. In this respect, we remark that for advection-diffusion problems the matrices of the discrete linear systems are non-symmetric.7.2Application to a System-EquationsWe use the DVS-framework to solve a Dirichlet boundary value problem, where displacements are zero over the boundary of the elastic body that occupies the domain Ω of the physical space.<a name="p306"></a> Over each one of such subdomains is solved a local problem by FEM, using linear functions as basis. On each node α of the mesh is defined a vector valued function u_;α with each component identified as uαi for i=1, 2, 3.<elsevierMultimedia ident="tbl0020"></elsevierMultimedia>Because our operators are symmetric and positive definite, we use CGM as an iterative procedure to solve those linear systems of equations that we have defined in the DVS framework.The code used in the previous section, which was originally developed to solve a single equation using finite differences, was adapted for solving systems of equations with FEM. We added the corresponding functionality in order to be able to solve systems of equations, in this case the elasticity problem.Example 5. A system of partial differential equations in three-dimensions has also been treated. This is the system of differential equations of static elasticity; namely:<a name="p307"></a><elsevierMultimedia ident="eq0355"></elsevierMultimedia>which was subject to the following Dirichlet boundary conditions:<elsevierMultimedia ident="eq0360"></elsevierMultimedia>The domain of study for our numerical experiments is a homogeneous isotropic linearly elastic unitary cube. In all of our experiments the primal nodes were located at edges of the subdomains, which is enough for A__t not being singular.We consider constant coefficients λ and μ equal to one. With these conditions we have a problem that has analytical solution, and is written as follows:<elsevierMultimedia ident="eq0365"></elsevierMultimedia>The <a class="elsevierStyleCrossRef" href="#tbl0025">Tables 5</a>, summarizes the numerical results obtained using the DVS methods with a tolerance of 10-7.8ConclusionsMathematical models of many geophysical systems lead to a great variety of partial differential equations (PDEs) whose solution methods are based on the computational processing of largescale algebraic systems [<a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Pinder, 2012</a>]. Parallel computing is outstanding among the new computational tools and, in order to effectively use the most advanced computers available today, massively parallel software is required. Domain decomposition methods (DDMs) have been developed precisely for effectively treating PDEs in parallel [<a class="elsevierStyleCrossRef" href="#bib0010">DDM Organization, 2012</a>]. What domain decomposition methods ideally intend to do has been summarized in this paper in the “DDM-paradigm”: to develop algorithms that ‘obtain the global solution by exclusively solving local problems’.In conclusion, in this paper:<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0055">1.We have presented a non-overlapping discretization method (the DVS-discretization) -in the sense that it uses a system of nodes such that each one of them belongs to one and only one subdomain of the coarse-mesh- applicable to a wide class of well-posed boundary problems associated with elliptic systems of equations. In particular, the differential operators may be symmetric, non-symmetric or indefinite (nonpositive-definite);</li><li class="elsevierStyleListItem" id="lsti0060">2.Four algorithms –the DVS-algorithms [<a class="elsevierStyleCrossRef" href="#bib0060">Herrera et al., 2011</a>]-, which were derived using the DVS-discretization and achieve the DDM-paradigm have been explained. Two of them are the result of using the BDDC and FETI-DP algorithms after applying DVS-discretization to the boundary value problem considered. The other two are obtained when two new algorithms, which had not been reported previously in the literature, were used instead;</li><li class="elsevierStyleListItem" id="lsti0065">3.Numerical procedures that permit achieving the DDM-paradigm with each one of the DVS-algorithms have been also presented;</li><li class="elsevierStyleListItem" id="lsti0070">4.Codes were developed and applied to several boundary values problems that occur in the modeling of certain geophysical phenomena, such as transport of solutes by both, free-fluids and fluids in a porous medium. We also present results for a static elasticity problem, which thereby illustrates the application of the algorithms to systems of differential equations; and</li><li class="elsevierStyleListItem" id="lsti0075">5.Besides their attractive parallelization properties, in the numerical examples the DVS-algorithms exhibited significantly improved numerical performance with respect to standard versions of BDDC and FETI-DP.</li></ul>" "textoCompletoSecciones" => array:1 [ "secciones" => array:14 [ 0 => array:3 [ "identificador" => "xres498039" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec519587" "titulo" => "Palabras clave" ] 2 => array:3 [ "identificador" => "xres498040" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 3 => array:2 [ "identificador" => "xpalclavsec519586" "titulo" => "Key words" ] 4 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 5 => array:2 [ "identificador" => "sec0010" "titulo" => "DVS Framework: A Summary" ] 6 => array:2 [ "identificador" => "sec0015" "titulo" => "The Derived-Vector Space (DVS)" ] 7 => array:2 [ "identificador" => "sec0020" "titulo" => "The Transformed Problem" ] 8 => array:3 [ "identificador" => "sec0025" "titulo" => "The DVS-Algorithms" "secciones" => array:2 [ 0 => array:3 [ "identificador" => "sec0030" "titulo" => "Primal Formulations" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0035" "titulo" => "The DVS Version of BDDC" ] 1 => array:2 [ "identificador" => "sec0040" "titulo" => "The DVS-Primal Algorithm" ] ] ] 1 => array:3 [ "identificador" => "sec0045" "titulo" => "Dual Formulations" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0050" "titulo" => "The DVS Version of FETI-DP" ] 1 => array:2 [ "identificador" => "sec0055" "titulo" => "The DVS-Dual Algorithm" ] ] ] ] ] 9 => array:3 [ "identificador" => "sec0060" "titulo" => "Numerical Procedures Fulfilling the DDM-Paradigm" "secciones" => array:4 [ 0 => array:2 [ "identificador" => "sec0065" "titulo" => "Comment on the DVS Numerical Procedures" ] 1 => array:2 [ "identificador" => "sec0070" "titulo" => "Application of S__" ] 2 => array:2 [ "identificador" => "sec0075" "titulo" => "Application of S__−1" ] 3 => array:2 [ "identificador" => "sec0080" "titulo" => "Application of a__ and j__." ] ] ] 10 => array:3 [ "identificador" => "sec0085" "titulo" => "Numerical Results" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0090" "titulo" => "Application of the DVS-algorithms to a Single-Equation" ] 1 => array:2 [ "identificador" => "sec0095" "titulo" => "Application to a System-Equations" ] ] ] 11 => array:2 [ "identificador" => "sec0100" "titulo" => "Conclusions" ] 12 => array:2 [ "identificador" => "xack161033" "titulo" => "Acknowledgement" ] 13 => array:1 [ "titulo" => "Bibliography" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2012-10-10" "fechaAceptado" => "2013-02-05" "PalabrasClave" => array:2 [ "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec519587" "palabras" => array:5 [ 0 => "computational-geophysics" 1 => "computational-PDEs" 2 => "non-overlapping DDM" 3 => "BDDC" 4 => "FETI-DP" ] ] ] "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Key words" "identificador" => "xpalclavsec519586" "palabras" => array:5 [ 0 => "computational-geophysics" 1 => "computational-PDEs" 2 => "non-overlapping DDM" 3 => "BDDC" 4 => "FETI-DP" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Los modelos matemáticos de muchos sistemas geofísicos requieren el procesamiento de sistemas algebraicos de gran escala. Las herramientas computacionales más avanzadas están masivamente paralelizadas. El software más efectivo para resolver ecuaciones diferenciales parciales en paralelo intenta alcanzar el paradigma de los métodos de descomposición de dominio, que hasta ahora se había mantenido como un anhelo no alcanzado. Sin embargo, un grupo de cuatro algoritmos –los algoritmos DVS- que lo alcanzan y que tiene aplicabilidad muy general se ha desarrollado recientemente. Este artículo está dedicado a presentarlos y a ilustrar su aplicación a problemas que se presentan frecuentemente en la investigación y el estudio de la Geofísica." ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "Mathematical models of many geophysical systems are based on the computational processing of large-scale algebraic systems. The most advanced computational tools are based on massively parallel processors. The most effective software for solving partial differential equations in parallel intends to achieve the DDM-paradigm. A set of four algorithms, the DVS-algorithms, which achieve it, and of very general applicability, has recently been developed and here they are explained. Also, their application to problems that frequently occur in Geophysics is illustrated." ] ] "NotaPie" => array:1 [ 0 => array:3 [ "etiqueta" => "1" "nota" => "In order to mimic standard notations, we should have used ∏ instead of the low-case ð. However, the modified definitions given here yield some convenient algebraic properties." "identificador" => "fn0005" ] ] "multimedia" => array:86 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 625 "Ancho" => 921 "Tamanyo" => 45042 ] ] "descripcion" => array:1 [ "en" => "The ‘original nodes’." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 619 "Ancho" => 910 "Tamanyo" => 56252 ] ] "descripcion" => array:1 [ "en" => "The original nodes in the coarse-mesh." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Figure 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 588 "Ancho" => 907 "Tamanyo" => 75082 ] ] "descripcion" => array:1 [ "en" => "The mitosis." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Figure 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 591 "Ancho" => 917 "Tamanyo" => 65732 ] ] "descripcion" => array:1 [ "en" => "The derived-nodes distributed in the coarse-mesh" ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 597 "Ancho" => 1337 "Tamanyo" => 171064 ] ] "descripcion" => array:1 [ "en" => "The numerical solution for the 2D case, here we use n=4." ] ] 5 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Figure 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 541 "Ancho" => 917 "Tamanyo" => 102079 ] ] "descripcion" => array:1 [ "en" => "The numerical solution for ν=0.01." ] ] 6 => array:7 [ "identificador" => "fig0040" "etiqueta" => "Figure 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr8.jpeg" "Alto" => 1341 "Ancho" => 1864 "Tamanyo" => 155764 ] ] "descripcion" => array:1 [ "en" => "Relative residual decay for the local mesh (16×16)." ] ] 7 => array:7 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETI-DP \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(2×2) × (2×2) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(4×4) × (4×4) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">16 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">225 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(6×6) × (6×6) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">36 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,225 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8) × (8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,969 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">49 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(10×10) × (10×10) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">100 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9,801 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">81 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(12×12) × (12×12) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">144 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">20,449 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(14×14) × (14×14) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">196 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">38,025 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">169 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(16×16) × (16×16) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">256 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">65,025 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">225 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(18×18) × (18×18) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">324 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">104,329 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">289 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(20×20) × (20×20) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">400 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">159,201 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">361 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(22×22) × (22×22) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">484 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">233,289 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">441 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(24×24) × (24×24) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">576 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">330,625 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">529 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(26×26) × (26×26) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">676 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">455,625 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">625 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(28×28) × (28×28) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">784 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">613,089 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">729 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(30×30) × (30×30) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">900 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">808,201 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">841 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795119.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Number of iterations made by the four DVS algorithms. The primal nodes were located at the vertices of subdomains." ] ] 8 => array:7 [ "identificador" => "tbl0010" "etiqueta" => "Table 2" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETI-DP \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(2×2×2) × (2×2×2) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">27 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(3×3×3) × (3×3×3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">27 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">80 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(4×4×4) × (4×4×4) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,375 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">351 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(5×5×5) × (5×5×5) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">125 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13,824 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(6×6×6) × (6×6×6) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">216 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">42,875 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">2,375 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(7×7×7) × (7×7×7) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">343 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">110,592 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,752 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8×8) × (8×8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">250,047 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8,575 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(9×9×9) × (9×9×9) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">729 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512,000 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14,336 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(10×10×10) × (10×10×10) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,000 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">970,299 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">22,599 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795116.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Number of iterations made by the four DVS algorithms. The primal nodes were located at edge." ] ] 9 => array:7 [ "identificador" => "tbl0015" "etiqueta" => "Table 3" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETI-DP \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">v \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8) × (64×64) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">49 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.01 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8) × (64×64) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">49 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8) × (64×64) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">49 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.0001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8) × (64×64) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">49 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.00001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(16×16) × (32×32) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">256 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">255 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">19 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">17 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">17 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">18 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">20 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.01 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(16×16) × (32×32) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">256 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">255 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">17 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(16×16) × (32×32) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">256 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">255 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">15 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.0001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(16×16) × (32×32) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">256 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">255 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">16 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.00001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(32×32) × (16×16) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">961 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">33 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">29 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">29 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">31 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">33 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.01 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(32×32) × (16×16) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">961 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">26 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">30 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(32×32) × (16×16) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">961 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">28 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.0001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(32×32) × (16×16) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">961 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">25 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">26 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">29 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.00001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(64×64) × (8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,096 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,969 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">53 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">52 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">53 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">59 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">52 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.01 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(64×64) × (8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,096 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,969 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">46 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">46 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">46 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">47 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">53 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.001 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(64×64) × (8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,096 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">261,121 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,969 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" 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title="table-entry " align="center" valign="middle">45 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">47 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">45 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">48 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">54 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="char" valign="middle">0.00001 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795117.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Comparison of the DVS-algorithms against the BDDC implemented in [<a class="elsevierStyleCrossRef" href="#bib0125">Mandel et al., 1996</a>]." ] ] 10 => array:7 [ "identificador" => "fig0030" "etiqueta" => "Figure 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true 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"mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "σππA__ΠΠ≡A__ππ−A__πIA__II−1A__πI" "Fichero" => "si149.jpeg" "Tamanyo" => 2714 "Alto" => 39 "Ancho" => 267 ] ] 58 => array:6 [ "identificador" => "eq0240" "etiqueta" => "(6.9)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A__II−1=∑α=1EA__IIα−1" "Fichero" => "si153.jpeg" "Tamanyo" => 2187 "Alto" => 54 "Ancho" => 156 ] ] 59 => array:6 [ "identificador" => "eq0245" "etiqueta" => "(6.10)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "v_;IA__II−1w_;I−A__Iπv_;π" "Fichero" => "si156.jpeg" "Tamanyo" => 1959 "Alto" => 33 "Ancho" => 166 ] ] 60 => array:6 [ "identificador" => "eq0250" "etiqueta" => "(6.11)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∑≡I∪Δ" "Fichero" => "si159.jpeg" "Tamanyo" => 752 "Alto" => 16 "Ancho" => 75 ] ] 61 => array:6 [ "identificador" => "eq0255" "etiqueta" => "(6.12)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∑∪π=Xand∑∩π=∅" "Fichero" => "si160.jpeg" "Tamanyo" => 1621 "Alto" => 16 "Ancho" => 175 ] ] 62 => array:6 [ "identificador" => "eq0260" "etiqueta" => "(6.13)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A__−1=A__−1ΠΠA__−1ΠΔA__−1ΔΠA__−1ΔΔ=A__−1∑∑A__−1∑πA__−1π∑A__−1ππ" "Fichero" => "si162.jpeg" "Tamanyo" => 7992 "Alto" => 71 "Ancho" => 411 ] ] 63 => array:6 [ "identificador" => "eq0265" "etiqueta" => "(6.14)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "S__−1=A__−1ΔΔ" "Fichero" => "si164.jpeg" "Tamanyo" => 1293 "Alto" => 30 "Ancho" => 115 ] ] 64 => array:6 [ "identificador" => "eq0270" "etiqueta" => "(6.15)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Wr∑=W∑" "Fichero" => "si165.jpeg" "Tamanyo" => 1346 "Alto" => 20 "Ancho" => 121 ] ] 65 => array:6 [ "identificador" => "eq0275" "etiqueta" => "(6.16)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "w_;≡A__−1v_;" "Fichero" => "si167.jpeg" "Tamanyo" => 806 "Alto" => 21 "Ancho" => 73 ] ] 66 => array:6 [ "identificador" => "eq0280" "etiqueta" => "(6.17)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "σππA__v_;π=w_;π−A__π∑A__∑∑t−1w_;∑,subjectedtoj__rv_;π=0" "Fichero" => "si169.jpeg" "Tamanyo" => 4419 "Alto" => 34 "Ancho" => 432 ] ] 67 => array:6 [ "identificador" => "eq0285" "etiqueta" => "(6.18)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "σππA__=A__ππ−A__π∑A__∑∑t−1A__∑π" "Fichero" => "si172.jpeg" "Tamanyo" => 2934 "Alto" => 34 "Ancho" => 272 ] ] 68 => array:6 [ "identificador" => "eq0290" "etiqueta" => "(6.19)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A__∑∑t−1=∑α=1EA__∑∑α−1" "Fichero" => "si173.jpeg" "Tamanyo" => 2644 "Alto" => 49 "Ancho" => 192 ] ] 69 => array:6 [ "identificador" => "eq0295" "etiqueta" => "(6.20)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "v_;∑=At__∑∑−1w_;∑−A__∑πv_;π" "Fichero" => "si177.jpeg" "Tamanyo" => 2755 "Alto" => 34 "Ancho" => 274 ] ] 70 => array:6 [ "identificador" => "eq0300" "etiqueta" => "(6.21)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A__−1ΔΔw_;=A__−1w_;ΔΔ" "Fichero" => "si180.jpeg" "Tamanyo" => 2059 "Alto" => 30 "Ancho" => 191 ] ] 71 => array:6 [ "identificador" => "eq0305" "etiqueta" => "(6.22)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "v_;≡A__−1w_;Δ" "Fichero" => "si183.jpeg" "Tamanyo" => 899 "Alto" => 21 "Ancho" => 95 ] ] 72 => array:6 [ "identificador" => "eq0310" "etiqueta" => "(6.23)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "v_;I+v_;Δ=v_;∑=−A__∑∑t−1A__∑πv_;π=−A__∑∑t−1A__∑πta__rv_;π." "Fichero" => "si184.jpeg" "Tamanyo" => 4579 "Alto" => 34 "Ancho" => 482 ] ] 73 => array:6 [ "identificador" => "eq0315" "etiqueta" => "(6.24)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "a__=a(i,α)(j,β)" "Fichero" => "si187.jpeg" "Tamanyo" => 1104 "Alto" => 21 "Ancho" => 98 ] ] 74 => array:6 [ "identificador" => "eq0320" "etiqueta" => "(6.25)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "a(i,α)(j,β)=1m(i)δij,∀α∈Z(i)and∀β∈Z(j)" "Fichero" => "si188.jpeg" "Tamanyo" => 2715 "Alto" => 24 "Ancho" => 292 ] ] 75 => array:6 [ "identificador" => "eq0325" "etiqueta" => "(6.26)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "j__w_;=w_;−a__w_;,foreveryw_;∈W" "Fichero" => "si190.jpeg" "Tamanyo" => 2067 "Alto" => 21 "Ancho" => 226 ] ] 76 => array:6 [ "identificador" => "eq0330" "etiqueta" => "(6.27)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "f_;Δ≡Rf⌢_;Δ−A__ΔΠA__ΠΠ−1Rf⌢_;Π" "Fichero" => "si193.jpeg" "Tamanyo" => 2729 "Alto" => 30 "Ancho" => 233 ] ] 77 => array:6 [ "identificador" => "eq0335" "etiqueta" => "(7.1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "−Δu=2π2n2sin(πnx)sin(πny),(x,y)∈[−1,1]×[−1,1],n=100u=0on∂Ω" "Fichero" => "si197.jpeg" "Tamanyo" => 4601 "Alto" => 18 "Ancho" => 548 ] ] 78 => array:6 [ "identificador" => "eq0340" "etiqueta" => "(7.2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "−Δu=3π2n2sin(πnx)sin(πny)sin(πnz),(x,y,z)∈[−1,1]×[−1,1]×[−1,1],n=100u=0on∂Ω" "Fichero" => "si198.jpeg" "Tamanyo" => 5872 "Alto" => 39 "Ancho" => 594 ] ] 79 => array:6 [ "identificador" => "eq0345" "etiqueta" => "(7.3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "−vΔu+b_;•∇u=0;(x,y)∈[0,1]×[0,1],b_;=(1,3)u(x,y)=0,(x,y)∈ψ11,(x,y)∈ψ2" "Fichero" => "si199.jpeg" "Tamanyo" => 5737 "Alto" => 71 "Ancho" => 365 ] ] 80 => array:6 [ "identificador" => "eq0350" "etiqueta" => "(7.4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "−vΔu+b_;•∇u=0;(x,y,z)∈[0,1]×[0,1],×[0,1],b_;=(1,1,1)u(x,y)=exp(x+y+z)on∂Ω" "Fichero" => "si200.jpeg" "Tamanyo" => 5496 "Alto" => 40 "Ancho" => 445 ] ] 81 => array:6 [ "identificador" => "eq0355" "etiqueta" => "(7.5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "(λ+μ)∇∇•u_;+μΔu_;=f_;Ω,iΩ" "Fichero" => "si202.jpeg" "Tamanyo" => 2065 "Alto" => 19 "Ancho" => 247 ] ] 82 => array:6 [ "identificador" => "eq0360" "etiqueta" => "(7.6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "u_;=0,on∂Ω" "Fichero" => "si203.jpeg" "Tamanyo" => 955 "Alto" => 14 "Ancho" => 122 ] ] 83 => array:6 [ "identificador" => "eq0365" "etiqueta" => "(7.7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "u_;=(sinπxsinπysinπz,sinπxsinπysinπz,sinπxsinπysinπz)" "Fichero" => "si205.jpeg" "Tamanyo" => 3880 "Alto" => 16 "Ancho" => 469 ] ] 84 => array:7 [ "identificador" => "tbl0020" "etiqueta" => "Table 4" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => false "mostrarDisplay" => true "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETI-DP \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(2×2×2) × (2×2×2) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">27 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(3×3×3) × (3×3×3) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">27 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">80 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">5 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(4×4×4) × (4×4×4) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">64 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">3,375 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">351 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">6 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(5×5×5) × (5×5×5) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">125 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13,824 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(6×6×6) × (6×6×6) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">216 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">42,875 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">2,375 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(7×7×7) × (7×7×7) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">343 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">110,592 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,752 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8×8) × (8×8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">250,047 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8,575 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">13 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(9×9×9) × (9×9×9) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">729 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512,000 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14,336 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">14 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(10×10×10) × (10×10×10) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,000 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">970,299 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">22,599 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">15 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795118.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Number of iterations made by the four DVS algorithms. The primal nodes were located at edges of the subdomains." ] ] 85 => array:7 [ "identificador" => "tbl0025" "etiqueta" => "Table 5" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => false "mostrarDisplay" => true "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETIDP \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(5×5×5) × (5×5×5) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">125 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">41,472 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">1,024 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">7 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">9 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(6×6×6) × (6×6×6) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">216 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">128,625 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">2,375 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">10 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(7×7×7) × (7×7×7) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">343 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">331,776 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">4,752 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">11 \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">(8×8×8) × (8×8×8) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">512 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">750,141 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8,575 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">8 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">12 \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795115.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Results for DVS Algorithms." ] ] ] "bibliografia" => array:2 [ "titulo" => "Bibliography" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0005" "bibliografiaReferencia" => array:33 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "Da Conceição,2006" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:1 [ "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "Da Conceição D.T. 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Antonio Carrillo-Ledesma, Ismael Herrera

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iherrera@geofsica.unam.mx

Corresponding author: Ismael Herrera

, Luis M. de la Cruz

Instituto de Geofísica Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México D.F., México

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          "en" => "<p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">The <span class="elsevierStyleItalic">original nodes</span> in the <span class="elsevierStyleItalic">coarse-mesh</span>&#46;</p>"
        ]
      ]
    ]
    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">1</span><span class="elsevierStyleSectionTitle" id="sect0025"><a name="p294"></a>Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">Mathematical models of many systems of interest&#44; including very important continuous systems of Earth Sciences and Engineering&#44; lead to a great variety of partial differential equations &#40;PDEs&#41; whose solution methods are based on the computational processing of large-scale algebraic systems&#46; Furthermore&#44; the incredible expansion experienced by the existing computational hardware and software has made amenable to effective treatment problems of an ever increasing diversity and complexity&#44; posed by scientific and engineering applications &#91;PITAC&#44; 2006&#93;&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">Parallel computing is outstanding among the new computational tools and&#44; in order to effectively use the most advanced computers available today&#44; massively parallel software is required&#46; Domain decomposition methods &#40;DDMs&#41; have been developed precisely for effectively treating PDEs in parallel &#91;<a class="elsevierStyleCrossRef" href="#bib0010">DDM Organization&#44; 2012</a>&#93;&#46; Ideally&#44; the main objective of domain decomposition research is to produce algorithms capable of &#8216;<span class="elsevierStyleItalic">obtaining the <span class="elsevierStyleUnderline">global</span> solution by exclusively solving <span class="elsevierStyleUnderline">local</span> problems</span>&#8217;&#44; but up-to-now this has only been an aspiration&#59; that is&#44; a strong desire for achieving such a property and so we call it &#8216;<span class="elsevierStyleItalic">the DDM-paradigm</span>&#8217;&#46; In recent times&#44; numerically competitive DDM-algorithms are <span class="elsevierStyleItalic">non-overlapping&#44; preconditioned</span> and necessarily incorporate <span class="elsevierStyleItalic">constraints</span> &#91;<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1991</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2000</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel&#44; 1993</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0140">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0115">J&#46; Li <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0165">Toselli <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#93;&#44; which pose an additional challenge for achieving the <span class="elsevierStyleItalic">DDM-paradigm</span>&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">Recently a group of four algorithms&#44; referred to as the &#8216;<span class="elsevierStyleItalic">DVS-algorithms</span>&#8217;&#44; which fulfill the <span class="elsevierStyleItalic">DDM-paradigm</span>&#44; was developed &#91;<a class="elsevierStyleCrossRef" href="#bib0050">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2012</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0050">L&#46;M&#46; de la Cruz <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2012</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0050">Herrera and L&#46;M&#46; de la Cruz <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2012</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Carrillo-Ledesma <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2012</a>&#93;&#46; To derive them a new discretization method&#44; which uses a non-overlapping system of nodes &#40;the <span class="elsevierStyleItalic">derived-nodes</span>&#41;&#44; was introduced&#46; This discretization procedure can be applied to any boundary-value problem&#44; or system of such equations&#46; In turn&#44; the resulting system of discrete equations can be treated using any available DDM-algorithm&#46; In particular&#44; two of the four <span class="elsevierStyleItalic">DVS-algorithms</span> mentioned above were obtained by application of the well-known and very effective algorithms BDDC and FETI-DP &#91;<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1991</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2000</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1993</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0140">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0115">J&#46; Li <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0165">Toselli <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#93;&#59; these will be referred to as the <span class="elsevierStyleItalic">DVS-BDDC</span> and <span class="elsevierStyleItalic">DVS-FETI-DP</span> algorithms&#46; The other two&#44; which will be referred to as the <span class="elsevierStyleItalic">DVS-PRIMAL</span> and <span class="elsevierStyleItalic">DVS-DUAL</span> algorithms&#44; were obtained by application of two new algorithms that had not been previously reported in the literature &#91;<a class="elsevierStyleCrossRef" href="#bib0060">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2011</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0070">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2010</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2009</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2009</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0080">Herrera&#44; 2008</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0085">Herrera&#44; 2007</a>&#93;&#46; As said before&#44; the four <span class="elsevierStyleItalic">DVS-algorithms</span> constitute a group of preconditioned and constrained algorithms that&#44; for the first time&#44; fulfill the <span class="elsevierStyleItalic">DDM-paradigm</span> &#91;<a class="elsevierStyleCrossRef" href="#bib0045">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2013</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0050">L&#46;M&#46; de la Cruz <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2012</a>&#93;&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">Both&#44; BDDC and FETI-DP&#44; are very well-known &#91;<a class="elsevierStyleCrossRef" href="#bib0025">Dohrmann&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0030">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1991</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2000</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0035">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0120">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1993</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0125">Mandel and Tezaur&#44; 1996</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0135">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2001</a>&#93;&#59; and both are highly efficient&#46; Recently&#44; it was established that these two methods are closely related and its numerical performance is quite similar &#91;<a class="elsevierStyleCrossRef" href="#bib0140">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2003</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0145">Mandel <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#93;&#46; On the other hand&#44; through numerical experiments&#44; we have established that the numerical performances of each one of the members of <span class="elsevierStyleItalic">DVS-algorithms</span> group &#40;<span class="elsevierStyleItalic">DVS-BDDC&#44; DVS-FETI-DP&#44; DVS-PRIMAL</span> and <span class="elsevierStyleItalic">DVS-DUAL</span>&#41; are very similar too&#46; Furthermore&#44; we have carried out comparisons of the performances of the standard versions of BDDC and FETI-DP with <span class="elsevierStyleItalic">DVS-BDDC</span> and <span class="elsevierStyleItalic">DVS-FETI-DP</span>&#44; and in all such numerical experiments the DVS algorithms have performed significantly better&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">Each <span class="elsevierStyleItalic">DVS-algorithm</span> possesses the following conspicuous features&#58;<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">&#8226;</span><p id="par0030" class="elsevierStylePara elsevierViewall">It fulfills the <span class="elsevierStyleItalic">DDM-paradigm</span>&#59;</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">&#8226;</span><p id="par0035" class="elsevierStylePara elsevierViewall">It is applicable to symmetric&#44; non-symmetric and indefinite matrices &#40;i&#46;e&#46;&#44; neither positive&#44; nor negative definite&#41;&#59; and</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">&#8226;</span><p id="par0040" class="elsevierStylePara elsevierViewall">It is preconditioned and constrained&#44; and has update numerical efficiency&#46;</p></li></ul></p><p id="par0045" class="elsevierStylePara elsevierViewall">Furthermore&#44; the uniformity of the algebraic structure of the matrix-formulas that define each one of them is remarkable&#46;</p><p id="par0050" class="elsevierStylePara elsevierViewall">This article is organized as follows&#46; In <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a> the basic definitions for the DVS framework are given&#59; here we define the set of &#8216;derived-nodes&#8217;&#44; internal&#44; interface&#44; primal and dual nodes&#44; the &#8216;derived-vector-space&#8217;&#44; among others&#46; <a class="elsevierStyleCrossRef" href="#sec0015">Section 3</a> is devoted to define the new set of vector spaces that conforms the DVS framework&#59; the Euclidean inner product&#44; is also defined here&#46; In <a class="elsevierStyleCrossRef" href="#sec0020">Section 4</a> the &#8216;transformed-problem&#8217; on the derived-nodes is explained in detail&#44; and this is our starting point to define the DVS algorithms&#46; <a class="elsevierStyleCrossRef" href="#sec0025">Section 5</a> presents a summary of the four DVS-algorithms&#58; DVS-BDDC&#44; DVS-FETI-DP&#44; DVS-PRIMAL and DVS-DUAL&#46; In <a class="elsevierStyleCrossRef" href="#sec0060">Section 6</a> we give the numerical procedures<a name="p295"></a> we use to fulfilling the DDM-paradigm&#44; and we explain in detail the implementation issues&#46; Finally&#44; in <a class="elsevierStyleCrossRef" href="#sec0085">Section 7</a> we show some numerical results obtained after the application of the DVS-algorithms in the solution of several boundary values problems of interest in Geophysics&#46; We studied examples for a single-equation&#44; for the cases of symmetric&#44; non-symmetric and indefinite problems&#46; We also present results for an elasticity problem&#44; where a system of PDE equations is solved&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2</span><span class="elsevierStyleSectionTitle" id="sect0030">DVS Framework&#58; A Summary</span><p id="par0055" class="elsevierStylePara elsevierViewall">The &#8216;<span class="elsevierStyleItalic">derived-vector-space framework</span> &#40;<span class="elsevierStyleItalic">DVS-framework</span>&#41;&#8217; is applied to the discrete system of equations that is obtained after the partial differential equation&#44; or system of such equations&#44; has been discretized&#46; The procedure is independent of the method of discretization that is used&#46; Thus&#44; the DVS-framework&#8217;s starting point is a system of linear algebraic equations that is referred to as the &#8216;<span class="elsevierStyleItalic">original problem</span>&#8217;&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0060" class="elsevierStylePara elsevierViewall">However&#44; in the <span class="elsevierStyleItalic">DVS</span> setting one does not work with the set of nodes originally used for discretizing the problem the <span class="elsevierStyleItalic">original-nodes</span>&#8217; &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#41;&#46; Instead&#44; one uses an auxiliary set of nodes&#58; the &#8216;<span class="elsevierStyleItalic">derived-nodes</span>&#8217;&#46; Each one of such nodes has the property that it belongs to one and only one subdomain of the <span class="elsevierStyleItalic">coarse mesh</span>&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0065" class="elsevierStylePara elsevierViewall">Indeed&#44; generally after a <span class="elsevierStyleItalic">coarse-mesh</span> has been introduced&#44; some <span class="elsevierStyleItalic">original-nodes</span> belong to more than one subdomain of the <span class="elsevierStyleItalic">coarse-mesh</span> &#40;<a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#41;&#44; which is inconvenient for achieving the <span class="elsevierStyleItalic">DDM-paradigm</span>&#46; Therefore&#44; in the <span class="elsevierStyleItalic">DVS-framework</span>&#44; each <span class="elsevierStyleItalic">original-node</span> that belongs to more than one subdomain is divided into as many new nodes &#8211; the <span class="elsevierStyleItalic">derived-nodes</span> &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>&#41; - as subdomains it belongs to&#46; Then&#44; the <span class="elsevierStyleItalic">derived-nodes</span> so obtained are distributed into the <span class="elsevierStyleItalic">coarse-mesh</span> subdomains so that each <span class="elsevierStyleItalic">derived-node</span> is assigned to one and only one subdomain of the <span class="elsevierStyleItalic">coarse-mesh</span> &#40;<a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>&#41;&#46; Once this has been done&#44; a convenient notation is to label each <span class="elsevierStyleItalic">derived-node</span> by a pair of natural numbers&#58; the first one indicating the <span class="elsevierStyleItalic">original-node</span> from which it derives and the second one&#44; the subdomain to which it is assigned&#46;<a name="p296"></a></p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><elsevierMultimedia ident="fig0015"></elsevierMultimedia><elsevierMultimedia ident="fig0020"></elsevierMultimedia><p id="par0070" class="elsevierStylePara elsevierViewall">The real-valued functions defined in the set of <span class="elsevierStyleItalic">derived-nodes</span> constitute a vector-space&#58; the &#8216;<span class="elsevierStyleItalic">derived-vector-space</span>&#8217;<span class="elsevierStyleItalic">&#44; W</span>&#46; This space becomes a finite-dimensional Hilbert-space when it is supplied with the inner-product that is usually introduced when dealing with real-valued functions defined in a set of nodes&#59; this is referred to as the <span class="elsevierStyleItalic">Euclidean inner-product</span>&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">Afterwards&#44; a new problem &#40;referred to as the &#8216;<span class="elsevierStyleItalic">transformed problem</span>&#8217;&#41; is defined in the <span class="elsevierStyleItalic">derived-vector-space</span>&#44; which is equivalent to the original system of discrete equations&#46; Thereafter&#44; all the numerical and computational work is carried out in the <span class="elsevierStyleItalic">DVS-space</span>&#46;</p><p id="par0080" class="elsevierStylePara elsevierViewall">Before leaving this Section&#44; we dwell a little further on the meaning of a <span class="elsevierStyleItalic">coarse-mesh</span>&#46; By it&#44; we mean a partition of &#937; into a set of non-overlapping subdomains &#123;&#937;<span class="elsevierStyleInf">1</span>&#44;&#46;&#46;&#46;&#44;&#937;<span class="elsevierStyleItalic"><span class="elsevierStyleInf">E</span></span>&#125;&#44; such that for each <span class="elsevierStyleItalic">&#945;</span>&#61;1&#44; &#46;&#46;&#46;&#44; <span class="elsevierStyleItalic">E</span>&#44; &#937;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#945;</span>&#8242;</span>&#44; is open and&#58;<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0085" class="elsevierStylePara elsevierViewall">Where &#937;&#175;&#945; stands for the closure of &#937;<span class="elsevierStyleItalic"><span class="elsevierStyleInf">&#945;</span></span>&#46; The set of &#8216;<span class="elsevierStyleItalic">subdomain-indices</span>&#8217; will be<elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0090" class="elsevierStylePara elsevierViewall">N&#710;&#945;&#44; <span class="elsevierStyleItalic">&#945;</span>&#61;1&#44;&#46;&#46;&#46;&#44; <span class="elsevierStyleItalic">E</span>&#44; will be used for the subset of <span class="elsevierStyleItalic">original-nodes</span> that correspond to nodes pertaining to &#937;&#175;&#945;&#46; As usual&#44; nodes will be classified into &#8216;<span class="elsevierStyleItalic">internal</span>&#8217; <span class="elsevierStyleItalic">and</span> &#8216;<span class="elsevierStyleItalic">interface-nodes</span>&#8217;&#58; a node is <span class="elsevierStyleItalic">internal</span> if it belongs to only one partition-subdomain closure and it is an <span class="elsevierStyleItalic">interface-node</span>&#44; when it belongs to more than one&#46; For the application of <span class="elsevierStyleItalic">dualprimal</span> methods&#44; <span class="elsevierStyleItalic">interface-nodes</span> are classified into &#8216;<span class="elsevierStyleItalic">primal</span>&#8217; and &#8216;<span class="elsevierStyleItalic">dual</span>&#8217; nodes&#46; We define&#58;</p><p id="par0100" class="elsevierStylePara elsevierViewall">N&#710;I&#8834;N&#710; as the set of <span class="elsevierStyleItalic">internal-nodes&#59;</span></p><p id="par0105" class="elsevierStylePara elsevierViewall">N&#710;&#915;&#8834;N&#710; as the set of <span class="elsevierStyleItalic">interface-nodes&#59;</span></p><p id="par0110" class="elsevierStylePara elsevierViewall">N&#710;&#960;&#8834;N&#710;&#915;&#8834;N&#710; as the set of <span class="elsevierStyleItalic">primal-nodes</span><a class="elsevierStyleCrossRef" href="#fn0005"><span class="elsevierStyleSup">1</span></a>&#59; and</p><p id="par0115" class="elsevierStylePara elsevierViewall">N&#710;&#916;&#8834;N&#710; as the set of <span class="elsevierStyleItalic">dual-nodes</span>&#46;</p><p id="par0120" class="elsevierStylePara elsevierViewall">The set of <span class="elsevierStyleItalic">primal-nodes</span> is required to be a subset of N&#710;&#915; and&#44; in principle&#44; could be otherwise chosen arbitrarily&#46; However&#44; the algorithms considered by <span class="elsevierStyleItalic">domain decomposition methods</span> are iterative-algorithms and their rate of convergence depends crucially on the selection of the set N&#710;&#960;&#46; Thus&#44; criteria for selecting N&#710;&#960; have been studied extensively &#40;see &#91;<a class="elsevierStyleCrossRef" href="#bib0165">Toselli <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#93;&#44; for detailed discussions of this topic&#41;&#46; Each one of the following two families of node-subsets is disjoint &#58;N&#710;I&#44;N&#710;&#915; and N&#710;I&#44;N&#710;&#960;&#44;N&#710;&#916;&#46; Furthermore&#44; these node subsets fulfill the relations&#58;<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0125" class="elsevierStylePara elsevierViewall">Throughout our developments the <span class="elsevierStyleItalic">original matrix</span>A&#8994;&#95;&#95; is assumed to be non-singular &#40;i&#46;e&#46;&#44; it defines a bijection of W&#710; into itself&#41;&#46; The following assumption &#40;&#8216;<span class="elsevierStyleItalic">axiom</span>&#8217;&#41; is also adopted in throughout the <span class="elsevierStyleItalic">DVS-framework</span>&#58; &#8220;When the indices p&#8712;N&#710;&#945; and q&#8712;N&#710;&#946; are <span class="elsevierStyleItalic">internal original-nodes</span>&#44; while <span class="elsevierStyleItalic">&#945; &#8800; &#946;</span>&#44; then p&#8712;N&#710;&#945; and q&#8712;N&#710;&#946; are unconnected&#8221;&#46; We recall that unconnected means&#58;<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3</span><span class="elsevierStyleSectionTitle" id="sect0035">The Derived-Vector Space &#40;DVS&#41;</span><p id="par0130" class="elsevierStylePara elsevierViewall">In order to have at hand a sufficiently general framework&#44; we consider functions defined on the set X of <span class="elsevierStyleItalic">derived-nodes</span> whose value at each <span class="elsevierStyleItalic">derived-node</span> is a <span class="elsevierStyleItalic">dD-Vector</span>&#46; The numerical applications that will be discussed in this paper correspond to two possible choices of <span class="elsevierStyleItalic">d</span>&#58; when the application refers to a single partial differential equation &#40;PDE&#41;&#44; <span class="elsevierStyleItalic">d</span>&#61;1&#44; and for the problems of elasticity that will be considered&#44; which are governed by a three-equations system&#44; <span class="elsevierStyleItalic">d</span>&#61;3&#46;</p><p id="par0135" class="elsevierStylePara elsevierViewall">Independently of the chosen value for <span class="elsevierStyleItalic">d</span>&#44; the set of such functions constitute a vector space&#44; <span class="elsevierStyleItalic">W</span>&#44; referred to as the &#8216;<span class="elsevierStyleItalic">derived-vector space</span>&#8217;&#46; When u&#95;&#59;&#8712;W&#44; we write <span class="elsevierStyleItalic">u</span>&#40;<span class="elsevierStyleItalic">p</span>&#44; <span class="elsevierStyleItalic">&#945;</span>&#41; for the value of u&#95;&#59; at the <span class="elsevierStyleItalic">derived-node</span> &#40;<span class="elsevierStyleItalic">p</span>&#44; &#945;&#41;&#46; We observe that&#44; in general&#44; <span class="elsevierStyleItalic">u</span>&#40;<span class="elsevierStyleItalic">p</span>&#44; <span class="elsevierStyleItalic">&#945;</span>&#41; itself is a <span class="elsevierStyleItalic">d-Vector</span> and we adopt the notation <span class="elsevierStyleItalic">u</span>&#40;<span class="elsevierStyleItalic">p</span>&#44; <span class="elsevierStyleItalic">&#945;&#44; i</span>&#41;&#44; <span class="elsevierStyleItalic">i</span>&#61;1&#44; &#46;&#46;&#46;&#44; <span class="elsevierStyleItalic">d</span>&#46; For the <span class="elsevierStyleItalic">i-th</span> component of <span class="elsevierStyleItalic">u</span>&#40;<span class="elsevierStyleItalic">p</span>&#44; <span class="elsevierStyleItalic">&#945;</span>&#41;&#46; When <span class="elsevierStyleItalic">d</span>&#61;1 the index <span class="elsevierStyleItalic">i</span> is irrelevant and&#44; in such a case&#44; will deleted throughout&#46;</p><p id="par0140" class="elsevierStylePara elsevierViewall">For every pair of functions&#44; u&#95;&#59;&#8712;W and w&#95;&#59;&#8712;W&#44; the &#8216;<span class="elsevierStyleItalic">Euclidean inner product</span>&#8217; is defined to be<a name="p297"></a></p><p id="par0145" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0150" class="elsevierStylePara elsevierViewall">Here&#44; u&#40;p&#44;&#945;&#41;&#8857;w&#40;p&#44;&#945;&#41; stands for the inner-product of the <span class="elsevierStyleItalic">dD-Vectors</span> involved&#59; thus&#44;<elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><p id="par0155" class="elsevierStylePara elsevierViewall">A fundamental property of the <span class="elsevierStyleItalic">derived-vector space W</span>&#44; is that it constitutes a finite dimensional <span class="elsevierStyleItalic">Hilbert-space</span> with respect to the <span class="elsevierStyleItalic">Euclidean innerproduct</span>&#46;</p><p id="par0160" class="elsevierStylePara elsevierViewall">Let W&#8242;&#8834;W be a linear subspace and assume M&#8834;X is a subset of <span class="elsevierStyleItalic">derived-nodes</span>&#46; Then&#44; the notation <span class="elsevierStyleItalic">W</span>&#8217;&#40;M&#41; will be used to represent the vector subspace of <span class="elsevierStyleItalic">W</span>&#8217;&#44; whose elements vanish at every <span class="elsevierStyleItalic">derived-node</span> that does not belong to M&#46; Furthermore&#44; corresponding to each <span class="elsevierStyleItalic">local subset of derived-nodes</span>&#44; X<span class="elsevierStyleItalic"><span class="elsevierStyleSup">&#945;</span></span>&#44; there is a &#8216;<span class="elsevierStyleItalic">local subspace of derived-vectors</span>&#8217;<span class="elsevierStyleItalic">&#44; W<span class="elsevierStyleSup">&#945;</span></span>&#44; which is defined by<elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0165" class="elsevierStylePara elsevierViewall">Clearly&#44; when u&#95;&#59;&#8712;W&#945;&#8834;W&#44; u&#95;&#59;&#40;p&#44;&#946;&#41;&#61;0 whenever <span class="elsevierStyleItalic">&#946;</span> &#8800; <span class="elsevierStyleItalic">&#945;</span>&#46; We observe that<elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0170" class="elsevierStylePara elsevierViewall">A derived-vector u&#95;&#59;&#8712;W is said to be <span class="elsevierStyleItalic">continuous</span> when u&#95;&#59;&#40;p&#44;&#945;&#41; is independent of <span class="elsevierStyleItalic">&#945;</span>&#46; The set of <span class="elsevierStyleItalic">continuous vectors</span> constitute the linear subspace&#44; <span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf">12</span>&#46;</p><p id="par0170a" class="elsevierStylePara elsevierViewall">The orthogonal complement &#40;with respect to the Euclidean inner-product&#41; of W12&#8834;W is W11&#8834;W&#46; Then W&#61;W11&#8853;W12&#46; Two projection-matrices a&#95;&#95;&#58;W&#8594;W and j&#95;&#95;&#58;W&#8594;W are here introduced&#59; they are the projection-operators&#44; with respect to the <span class="elsevierStyleItalic">Euclidean inner-product</span> on <span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf">12</span> and <span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf">11</span>&#44; respectively&#46; When u&#95;&#59;&#8712;W&#44; one has<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0175" class="elsevierStylePara elsevierViewall">the vectors j&#95;&#95;u&#95;&#59; and a&#95;&#95;u&#95;&#59; are said to be the &#8216;<span class="elsevierStyleItalic">jump</span>&#8217; and the &#8216;<span class="elsevierStyleItalic">average</span>&#8217; of u&#95;&#59;&#44; respectively&#46; Therefore&#44; <span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf">11</span> is the &#8216;<span class="elsevierStyleItalic">zero-average</span>&#8217; subspace&#44; while <span class="elsevierStyleItalic">W</span><span class="elsevierStyleInf">12</span> is the &#8216;<span class="elsevierStyleItalic">zero-jump</span>&#8217; subspace&#46;</p><p id="par0180" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Original-nodes</span> are classified into &#8216;<span class="elsevierStyleItalic">internal</span>&#8217; and &#8216;<span class="elsevierStyleItalic">interface-nodes</span>&#8217;&#58; a node is <span class="elsevierStyleItalic">internal</span> if it belongs to only one subdomain-closure of the <span class="elsevierStyleItalic">coarse-mesh</span>&#44; and it is an <span class="elsevierStyleItalic">interface-node</span> when it belongs to more than one of such closure-subdomains&#46; Some subspaces&#44; significant for our developments&#44; are listed next&#58;<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0185" class="elsevierStylePara elsevierViewall">At present&#44; numerically competitive algorithms need to incorporate <span class="elsevierStyleItalic">restrictions</span> and to this end&#44; in the <span class="elsevierStyleItalic">DVS-framework</span>&#44; a &#8216;<span class="elsevierStyleItalic">restricted subspace</span>&#8217; Wr&#8834;W is selected&#46; In the developments that follow&#44; it is assumed that&#58;<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0190" class="elsevierStylePara elsevierViewall">The matrix a&#95;&#95;r will be the projection-operator on <span class="elsevierStyleItalic">W<span class="elsevierStyleInf">r</span></span>&#46; We observe that when u&#95;&#59;&#8712;&#40;WI&#43;W&#916;&#41;&#44; one has a&#95;&#95;ru&#95;&#59;&#61;u&#95;&#59;&#46; We also notice that<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">4</span><span class="elsevierStyleSectionTitle" id="sect0040">The Transformed Problem</span><p id="par0195" class="elsevierStylePara elsevierViewall">The <span class="elsevierStyleItalic">transformed-problem</span> consists in finding u&#95;&#59;&#8712;W such that<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0200" class="elsevierStylePara elsevierViewall">Where&#58;<elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0205" class="elsevierStylePara elsevierViewall">and<elsevierMultimedia ident="eq0080"></elsevierMultimedia></p><p id="par0210" class="elsevierStylePara elsevierViewall">together with<elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0215" class="elsevierStylePara elsevierViewall">The function <span class="elsevierStyleItalic">m</span> &#40;<span class="elsevierStyleItalic">p&#44; q</span>&#41; is said to be the &#8216;<span class="elsevierStyleItalic">multiplicity</span>&#8217; of the pair &#40;<span class="elsevierStyleItalic">p&#44; q</span>&#41;&#46; The &#8216;<span class="elsevierStyleItalic">derived-nodes</span>&#8217; are created after a <span class="elsevierStyleItalic">coarse-mesh</span> has been<a name="p298"></a> introduced&#44; by dividing the <span class="elsevierStyleItalic">original-nodes</span> as explained in the Overview &#40;<a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>&#41;&#44; and then with each &#8216;<span class="elsevierStyleItalic">derived-node</span>&#8217; we associate a unique pair of numbers &#40;<span class="elsevierStyleItalic">p&#44; &#945;</span>&#41; such that &#945;&#8712;E&#710; and p&#8712;N&#710;&#945;&#46; In what follows&#44; we identify <span class="elsevierStyleItalic">derived-nodes</span> with such pairs&#46;</p><p id="par0220" class="elsevierStylePara elsevierViewall">Then&#44; in order to incorporate the constraints&#44; we define<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0225" class="elsevierStylePara elsevierViewall">then&#44; the matrix A&#95;&#95;&#58;Wr&#8594;Wr defined by<elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0230" class="elsevierStylePara elsevierViewall">has the property that<elsevierMultimedia ident="eq0100"></elsevierMultimedia></p><p id="par0235" class="elsevierStylePara elsevierViewall">Hence&#44; <a class="elsevierStyleCrossRef" href="#eq0070">Eq&#46; &#40;4&#46;1&#41;</a> is replaced by<elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0240" class="elsevierStylePara elsevierViewall">For matrices and vectors the following notation is adopted&#58;<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0245" class="elsevierStylePara elsevierViewall">where the matrices<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0250" class="elsevierStylePara elsevierViewall">furthermore&#44;<elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0255" class="elsevierStylePara elsevierViewall">The matrix A&#95;&#95;&#58;W&#8594;W will be referred to as the &#8216;<span class="elsevierStyleItalic">transformed-matrix</span>&#8217;&#46; We observe that A&#95;&#95;&#61;A&#95;&#95;t when <span class="elsevierStyleItalic">&#960;</span> &#61; &#216;&#46;</p><p id="par0260" class="elsevierStylePara elsevierViewall">In turn&#44; the <span class="elsevierStyleItalic">transformed problem</span> of &#40;4&#46;8&#41; can be reduced&#44; see &#91;<a class="elsevierStyleCrossRef" href="#bib0070">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2010</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0065">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2009</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0080">Herrera&#44; 2008</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0085">Herrera&#44; 2007</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0040">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2000</a>&#93; for details&#44; into the following problem&#44; which is expressed in terms of the values of the solution at <span class="elsevierStyleItalic">dual-nodes</span>&#44; exclusively&#58; &#8220;Find u&#95;&#59;&#916;&#8712;W &#40;&#8710;&#41; that satisfies<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0265" class="elsevierStylePara elsevierViewall">Here&#44; f&#95;&#59;&#916;&#8712;a&#95;&#95;W&#40;&#916;&#41; and the &#8216;<span class="elsevierStyleItalic">Schur-complement matrix with constraints</span>&#8217; are defined by<elsevierMultimedia ident="eq0130"></elsevierMultimedia></p><p id="par0270" class="elsevierStylePara elsevierViewall">and<elsevierMultimedia ident="eq0135"></elsevierMultimedia></p><p id="par0275" class="elsevierStylePara elsevierViewall">respectively&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5</span><span class="elsevierStyleSectionTitle" id="sect0045">The DVS-Algorithms</span><p id="par0280" class="elsevierStylePara elsevierViewall">Generally two kinds of approaches are distinguished&#58; primal &#8211;these are direct approaches&#44; which do not resort to Lagrange multipliers- and dual &#8211;indirect approaches that use Lagrange multipliers-&#46; However&#44; when DDMs are formulated using a setting as general as that supplied by the <span class="elsevierStyleItalic">DVS-framework</span>&#44; such a distinction is irrelevant&#46; The feature that is conspicuous for different options is the information that the algorithm seeks&#46; Indeed&#44; four algorithms will be obtained by seeking successively for the vectors&#58; u&#95;&#59;&#916;&#44;S&#95;&#95;&#8722;1j&#95;&#95;S&#95;&#95;u&#95;&#59;&#916;&#44; j&#95;&#95;S&#95;&#95;u&#95;&#59;&#916;&#44; and S&#95;&#95;u&#95;&#59;&#916;&#46; However&#44; in the presentation that follows we stick to the &#8216;<span class="elsevierStyleItalic">primal vs&#46; dual-algorithms</span>&#8217; classification&#46;</p><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0050">Primal Formulations</span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">The DVS Version of BDDC</span><p id="par0285" class="elsevierStylePara elsevierViewall">This is a primal algorithm which seeks directly for u&#95;&#59;&#916;&#46; Pre-multiplying <a class="elsevierStyleCrossRef" href="#eq0125">Eq&#46; &#40;4&#46;12&#41;</a> by a&#95;&#95;S&#95;&#95;&#8722;1&#44; one gets&#58;<elsevierMultimedia ident="eq0140"></elsevierMultimedia></p><p id="par0290" class="elsevierStylePara elsevierViewall">In &#91;<a class="elsevierStyleCrossRef" href="#bib0040">Farhat <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2000</a>&#93;&#44; it was shown that <a class="elsevierStyleCrossRef" href="#eq0140">Eq&#46; &#40;5&#46;1&#41;</a> is equivalent to <a class="elsevierStyleCrossRef" href="#eq0125">Eq&#46; &#40;4&#46;12&#41;</a>&#46; This equation is the DVS-version of BDDC&#46;</p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">The DVS-Primal Algorithm</span><p id="par0295" class="elsevierStylePara elsevierViewall">For this algorithm&#44; the <span class="elsevierStyleItalic">sought-information</span> is&#58;<a name="p299"></a><elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0305" class="elsevierStylePara elsevierViewall">Applying to a&#95;&#95;S&#95;&#95;<a class="elsevierStyleCrossRef" href="#eq0145">Eq&#46; &#40;5&#46;2&#41;</a> it is seen that a&#95;&#95;S&#95;&#95;v&#95;&#59;&#916;&#61;0&#46; Furthermore&#44;<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0310" class="elsevierStylePara elsevierViewall">Therefore<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0315" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0155">Eq&#46; &#40;5&#46;4&#41;</a> does not define an iterative algorithm&#46; In order to obtain such an algorithm&#44; we project on j&#95;&#95;S&#95;&#95;&#8722;1W&#916;&#44; to obtain&#58;<elsevierMultimedia ident="eq0160"></elsevierMultimedia></p><p id="par0320" class="elsevierStylePara elsevierViewall">This algorithm is referred to as the &#8216;<span class="elsevierStyleItalic">DVS-primal algorithm</span>&#8217;&#46; The solution is given by<elsevierMultimedia ident="eq0165"></elsevierMultimedia></p><p id="par0325" class="elsevierStylePara elsevierViewall">We observe that we could have written u&#95;&#59;&#916;&#61;v&#95;&#59;&#916;&#43;S&#95;&#95;&#8722;1f&#95;&#59;&#916; instead of <a class="elsevierStyleCrossRef" href="#eq0165">Eq&#46; &#40;5&#46;6&#41;</a>&#46; However&#44; the application of the projection operator a&#95;&#95; is important when &#957;&#95;&#59;&#916; and S&#95;&#95;&#8722;1f&#95;&#59;&#916; are not computed with exact arithmetic&#44; as it is the case when using numerical methods&#44; because when it is applied it replaces v&#95;&#59;&#916;&#43;S&#95;&#95;&#8722;1f&#95;&#59;&#916; by the continuous-vector closest &#40;with respect to the Euclidean distance&#41; to it&#46;</p></span></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0065">Dual Formulations</span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">The DVS Version of FETI-DP</span><p id="par0330" class="elsevierStylePara elsevierViewall">For this algorithm the <span class="elsevierStyleItalic">sought-information</span> is defined to be&#58; &#955;&#95;&#59;&#916;&#8801;&#8722;j&#95;&#95;S&#95;&#95;u&#95;&#59;&#916;&#46; This algorithm can be easily derived from the <span class="elsevierStyleItalic">DVS-primal</span> formulation that has just been presented&#46; We observe that v&#95;&#59;&#916;&#61;S&#95;&#59;&#8722;1&#955;&#95;&#59;&#916;&#44;&#955;&#95;&#59;&#916;&#61;S&#95;&#95;&#957;&#95;&#59;&#916;&#44; in view of <a class="elsevierStyleCrossRef" href="#eq0145">Eq&#46; &#40;5&#46;2&#41;</a>&#44; and a&#95;&#95;&#955;&#95;&#59;&#916;&#61;0&#46; This permits transforming <a class="elsevierStyleCrossRef" href="#eq0160">Eq&#46; &#40;5&#46;5&#41;</a> into<elsevierMultimedia ident="eq0170"></elsevierMultimedia></p><p id="par0335" class="elsevierStylePara elsevierViewall">Applying S&#95;&#95;&#8722;1 to the first of these equations&#44; it is obtained&#58;<elsevierMultimedia ident="eq0175"></elsevierMultimedia></p><p id="par0340" class="elsevierStylePara elsevierViewall">As for <a class="elsevierStyleCrossRef" href="#eq0165">Eq&#46; &#40;5&#46;6&#41;</a>&#44; it becomes&#58;<elsevierMultimedia ident="eq0180"></elsevierMultimedia></p></span><span id="sec0055" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0075">The DVS-Dual Algorithm</span><p id="par0345" class="elsevierStylePara elsevierViewall">In this algorithm&#44; the <span class="elsevierStyleItalic">sought-information</span> is&#58; &#956;&#95;&#59;&#916;&#8801;S&#95;&#95;u&#95;&#59;&#916;&#46; Then&#44; u&#95;&#59;&#916;&#61;S&#95;&#95;&#8722;1&#956;&#95;&#59;&#916;&#46; Replacing this in <a class="elsevierStyleCrossRef" href="#eq0140">Eq&#46; &#40;5&#46;1&#41;</a>&#44; one gets&#58;<elsevierMultimedia ident="eq0185"></elsevierMultimedia></p><p id="par0350" class="elsevierStylePara elsevierViewall">Finally&#44; multiplying by S&#95;&#95; the first of these equalities&#44; it is obtained&#58;<elsevierMultimedia ident="eq0190"></elsevierMultimedia></p><p id="par0355" class="elsevierStylePara elsevierViewall">When &#956;&#95;&#59;&#916; is known&#44; u&#95;&#59;&#916; can be recovered by means of<elsevierMultimedia ident="eq0195"></elsevierMultimedia></p><p id="par0360" class="elsevierStylePara elsevierViewall">A comment similar to that made immediately after <a class="elsevierStyleCrossRef" href="#eq0165">Eq&#46; &#40;5&#46;6&#41;</a>&#44; goes here&#58; we have applied the projection matrix a&#95;&#95;&#44; in <a class="elsevierStyleCrossRef" href="#eq0195">Eq&#46; &#40;5&#46;12&#41;</a> because we are assuming that exact arithmetic generally will not be used&#46;</p></span></span></span><span id="sec0060" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6</span><span class="elsevierStyleSectionTitle" id="sect0080">Numerical Procedures Fulfilling the DDM-Paradigm</span><p id="par0365" class="elsevierStylePara elsevierViewall">Summarizing&#44; the preconditioned <span class="elsevierStyleItalic">DVS-algorithms</span> with constraints are&#58; <a name="p300"></a><elsevierMultimedia ident="eq0200"></elsevierMultimedia><elsevierMultimedia ident="eq0205"></elsevierMultimedia><elsevierMultimedia ident="eq0210"></elsevierMultimedia><elsevierMultimedia ident="eq0215"></elsevierMultimedia></p><span id="sec0065" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0085">Comment on the DVS Numerical Procedures</span><p id="par0370" class="elsevierStylePara elsevierViewall">The outstanding uniformity of the formulas given in <a class="elsevierStyleCrossRef" href="#eq0200">Eqs&#46; &#40;6&#46;1&#41;</a> to <a class="elsevierStyleCrossRef" href="#eq0215">&#40;6&#46;4&#41;</a> yields clear advantages for code development&#44; especially when such codes are built using object-oriented programming techniques&#46; Such advantages include&#58;<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0020"><span class="elsevierStyleLabel">I&#46;</span><p id="par0375" class="elsevierStylePara elsevierViewall">The construction of very robust codes&#46; This is an advantage of the <span class="elsevierStyleItalic">DVS-algorithms</span>&#44; which stems from the fact the definitions of such algorithms exclusively depend on the discretized system of equations&#44; obtained after discretization of the partial differential equations considered &#40;referred to as the <span class="elsevierStyleItalic">original problem</span>&#41;&#44; but which is otherwise independent of the problem that motivated it&#46; In this manner&#44; for example&#44; essentially the same code was applied to treat 2-D and 3-D problems&#59; indeed&#44; only the part defining the geometry had to be changed&#44; and that was a very small part of it&#59;</p></li><li class="elsevierStyleListItem" id="lsti0025"><span class="elsevierStyleLabel">II&#46;</span><p id="par0380" class="elsevierStylePara elsevierViewall">The codes may use different local solvers&#44; which can be direct or iterative solvers&#59;</p></li><li class="elsevierStyleListItem" id="lsti0030"><span class="elsevierStyleLabel">III&#46;</span><p id="par0385" class="elsevierStylePara elsevierViewall">Minimal modifications are required for transforming sequential codes into parallel ones&#59; and</p></li><li class="elsevierStyleListItem" id="lsti0035"><span class="elsevierStyleLabel">IV&#46;</span><p id="par0390" class="elsevierStylePara elsevierViewall">Such formulas also permit developing codes which fulfill the <span class="elsevierStyleItalic">DDM-paradigm</span>&#59; i&#46;e&#46;&#44; in which &#8220;the solution of the <span class="elsevierStyleUnderline">global</span> problem is obtained by exclusively solving <span class="elsevierStyleUnderline">local</span> problems&#8221;&#46;</p></li></ul></p><p id="par0395" class="elsevierStylePara elsevierViewall">This last property makes the DVS-algorithms very suitable as a tool to be used in the construction of massively-parallelized software&#44; so much needed for efficiently programming the most powerful parallel computers available at present&#46; In the next Subsection&#44; procedures for constructing codes possessing Property IV are explained with some detail&#46;</p><p id="par0400" class="elsevierStylePara elsevierViewall">All the DVS-algorithms of <a class="elsevierStyleCrossRef" href="#eq0200">Eqs&#46; &#40;6&#46;1&#41;</a> to <a class="elsevierStyleCrossRef" href="#eq0215">&#40;6&#46;4&#41;</a> are iterative and can be implemented with recourse to Conjugate Gradient Method &#40;CGM&#41;&#44; when the matrix is definite and symmetric&#44; or some other iterative procedure such as GMRES&#44; when that is not the case&#46; At each iteration step&#44; depending on the <span class="elsevierStyleItalic">DVS-algorithm</span> that is applied&#44; one has to compute the action on a <span class="elsevierStyleItalic">derived-vector</span> of one of the following matrices&#58; a&#95;&#95;S&#95;&#95;&#8722;1a&#95;&#95;S&#95;&#95;&#44; j&#95;&#95;S&#95;&#95;j&#95;&#95;S&#95;&#95;&#8722;1&#44; S&#95;&#95;&#8722;1j&#95;&#95;S&#95;&#95;j&#95;&#95; or S&#95;&#95;a&#95;&#95;S&#95;&#95;&#8722;1a&#95;&#95;&#46; Such matrices in turn are different permutations of the matrices S&#95;&#95;&#44; S&#95;&#95;&#8722;1&#44; a&#95;&#95; and J&#95;&#95;&#46; Thus&#44; to implement any of the preconditioned <span class="elsevierStyleItalic">DVS-algorithms</span>&#44; one only needs to separately develop codes capable of computing the action of each one of the matrices S&#95;&#95;&#44; S&#95;&#95;&#8722;1&#44; a&#95;&#95; or j&#95;&#95; on an arbitrary <span class="elsevierStyleItalic">derived-vector</span>&#44; of <span class="elsevierStyleItalic">W</span>&#46;</p><p id="par0405" class="elsevierStylePara elsevierViewall">Therefore&#44; next we present numerical procedures for computing the application of each one of the matrices S&#95;&#95;&#44; S&#95;&#95;&#8722;1&#44; a&#95;&#95; and j&#95;&#95; which fulfill the <span class="elsevierStyleItalic">DDM-paradigm</span>&#46; It will be seen that only a&#95;&#95; requires exchange of information between derived-nodes belonging to different subdomains&#59; actually&#44; between <span class="elsevierStyleItalic">derived-nodes</span> that are descendants of the same <span class="elsevierStyleItalic">original-node</span> &#40;the exchange of information is minimal&#41;&#46; As for j&#95;&#95;&#61;I&#95;&#95;&#8722;a&#95;&#95; once the action of a&#95;&#95; has been computed&#44; no further exchange of information is required&#46;</p></span><span id="sec0070" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0090">Application of S&#95;&#95;</span><p id="par0410" class="elsevierStylePara elsevierViewall">From <a class="elsevierStyleCrossRef" href="#eq0130">Eq&#46; &#40;4&#46;13&#41;</a>&#44; we recall the definition of the matrix S&#95;&#95;&#8801;A&#95;&#95;&#916;&#916;&#8722;A&#95;&#95;&#916;&#928;A&#95;&#95;&#928;&#928;&#8722;1A&#95;&#95;&#928;&#916;&#46; In order to evaluate the action of S&#95;&#95; on any <span class="elsevierStyleItalic">derived-vector</span>&#44; we need to successively evaluate the action of the following matrices A&#95;&#95;&#928;&#916;&#44; A&#95;&#95;&#928;&#928;&#8722;1&#44; A&#95;&#95;&#916;&#928; and A&#95;&#95;&#916;&#916;&#46; Nothing special is required except for A&#95;&#95;&#928;&#928;&#8722;1&#46; A procedure for evaluating the action of this matrix&#44; which fulfills the <span class="elsevierStyleItalic">DDM-paradigm</span> is explained next&#46;</p><p id="par0415" class="elsevierStylePara elsevierViewall">We have<elsevierMultimedia ident="eq0220"></elsevierMultimedia></p><p id="par0420" class="elsevierStylePara elsevierViewall">Let &#957;&#95;&#59;&#8712;W&#44; be an arbitrary <span class="elsevierStyleItalic">derived-vector</span>&#44; and write<elsevierMultimedia ident="eq0225"></elsevierMultimedia></p><p id="par0425" class="elsevierStylePara elsevierViewall">Then&#44; w&#95;&#59;&#61;w&#95;&#59;I&#43;w&#95;&#59;&#960;&#8712;W is characterized by<elsevierMultimedia ident="eq0230"></elsevierMultimedia></p><p id="par0430" class="elsevierStylePara elsevierViewall">and can obtained iteratively&#46; Here&#44;<a name="p301"></a><elsevierMultimedia ident="eq0235"></elsevierMultimedia></p><p id="par0435" class="elsevierStylePara elsevierViewall">and&#44; with a&#95;&#95;&#960; as the projection-matrix into Wr&#40;&#960;&#41;&#44;j&#95;&#95;&#960;&#8801;I&#95;&#95;&#8722;a&#95;&#95;&#960;&#46;</p><p id="par0440" class="elsevierStylePara elsevierViewall">We observe that fulfilling the <span class="elsevierStyleItalic">DDM-paradigm</span> when computing the action of A&#95;&#95;II&#8722;1 is straightforward because<elsevierMultimedia ident="eq0240"></elsevierMultimedia></p><p id="par0445" class="elsevierStylePara elsevierViewall">is parallelizable&#46; Once v&#95;&#59;&#960;&#8712;Wr&#40;&#960;&#41; has been obtained&#44; to derive &#957;&#95;&#59;I one can apply&#58;<elsevierMultimedia ident="eq0245"></elsevierMultimedia></p><p id="par0450" class="elsevierStylePara elsevierViewall">this completes the evaluation of S&#95;&#95;&#46;</p></span><span id="sec0075" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;3</span><span class="elsevierStyleSectionTitle" id="sect0095">Application of S&#95;&#95;&#8722;1</span><p id="par0455" class="elsevierStylePara elsevierViewall">We define<elsevierMultimedia ident="eq0250"></elsevierMultimedia></p><p id="par0460" class="elsevierStylePara elsevierViewall">and observe that<elsevierMultimedia ident="eq0255"></elsevierMultimedia></p><p id="par0465" class="elsevierStylePara elsevierViewall">Therefore&#44; the matrix A&#95;&#95;&#8722;1 can be written as&#58;<elsevierMultimedia ident="eq0260"></elsevierMultimedia></p><p id="par0470" class="elsevierStylePara elsevierViewall">Furthermore&#44; S&#95;&#95;&#58;W&#916;&#8594;W&#916; fulfills<elsevierMultimedia ident="eq0265"></elsevierMultimedia></p><p id="par0475" class="elsevierStylePara elsevierViewall">Another property that is relevant for the following discussion is&#58;<elsevierMultimedia ident="eq0270"></elsevierMultimedia></p><p id="par0480" class="elsevierStylePara elsevierViewall">for any &#957;&#95;&#59;&#8712;W&#44; let us write<elsevierMultimedia ident="eq0275"></elsevierMultimedia></p><p id="par0485" class="elsevierStylePara elsevierViewall">then&#44; w&#95;&#59;&#960; fulfills<elsevierMultimedia ident="eq0280"></elsevierMultimedia></p><p id="par0490" class="elsevierStylePara elsevierViewall">Here&#44; j&#95;&#95;r&#8801;I&#95;&#95;&#8722;a&#95;&#95;r&#44; where the matrix a&#95;&#95;r is the projection operator on <span class="elsevierStyleItalic">W<span class="elsevierStyleInf">r</span></span><span class="elsevierStyleInf">&#8242;</span> while<elsevierMultimedia ident="eq0285"></elsevierMultimedia></p><p id="par0495" class="elsevierStylePara elsevierViewall">Furthermore&#44; we observe that<elsevierMultimedia ident="eq0290"></elsevierMultimedia></p><p id="par0500" class="elsevierStylePara elsevierViewall">In order to use <a class="elsevierStyleCrossRef" href="#eq0290">Eq&#46; &#40;6&#46;19&#41;</a> as a means of parallelizing the DVS-algorithms&#44; however&#44; the detailed discussion of such procedures will be presented separately &#91;<a class="elsevierStyleCrossRef" href="#bib0045">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2013</a>&#59; L&#46;M&#46; de la Cruz <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2013&#93;&#46; It is necessary that the local matrices&#44; A&#95;&#95;&#8721;&#8721;&#945;&#44; be invertible&#46; This is granted when invertible A&#95;&#95; in <span class="elsevierStyleItalic">W<span class="elsevierStyleInf">r</span></span><span class="elsevierStyleInf">&#8242;</span> which generally is achieved by taking a sufficiently large number of <span class="elsevierStyleItalic">primal-nodes</span>&#46;</p><p id="par0505" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0280">Eq&#46; &#40;6&#46;17&#41;</a> is solved iteratively&#46; Once &#957;&#95;&#59;&#960; has been obtained&#44; we apply&#58;<elsevierMultimedia ident="eq0295"></elsevierMultimedia></p><p id="par0510" class="elsevierStylePara elsevierViewall">This procedure permits obtaining A&#95;&#95;&#8722;1w&#95;&#59; in full&#59; however&#44; we only need A&#95;&#95;&#8722;1&#916;&#916;w&#95;&#59;&#46; We observe that<elsevierMultimedia ident="eq0300"></elsevierMultimedia></p><p id="par0515" class="elsevierStylePara elsevierViewall">The vector A&#95;&#95;&#8722;1w&#95;&#59;&#916; can be obtained by the general procedure presented above&#46; Thus&#44; take w&#95;&#59;&#8801;w&#95;&#59;&#916;&#8712;W&#916;&#8834;W and<elsevierMultimedia ident="eq0305"></elsevierMultimedia></p><p id="par0520" class="elsevierStylePara elsevierViewall">Therefore&#44; <a name="p302"></a><elsevierMultimedia ident="eq0310"></elsevierMultimedia></p></span><span id="sec0080" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;4</span><span class="elsevierStyleSectionTitle" id="sect0100">Application of a&#95;&#95; and j&#95;&#95;&#46;</span><p id="par0525" class="elsevierStylePara elsevierViewall">We use the notation<elsevierMultimedia ident="eq0315"></elsevierMultimedia></p><p id="par0530" class="elsevierStylePara elsevierViewall">then &#91;<a class="elsevierStyleCrossRef" href="#bib0070">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2010</a>&#93;&#58;<elsevierMultimedia ident="eq0320"></elsevierMultimedia></p><p id="par0535" class="elsevierStylePara elsevierViewall">while j&#95;&#95;&#61;I&#95;&#95;&#8722;a&#95;&#95; therefore&#44;<elsevierMultimedia ident="eq0325"></elsevierMultimedia></p><p id="par0540" class="elsevierStylePara elsevierViewall">Therefore&#44; only the evaluation of a&#95;&#95;u&#95;&#59; requires exchange of information between subdomains&#46; In general&#44; such numbers are very small&#59; for example in application to single-equation problem&#44; when an orthogonal grid is used&#44; they are at most&#58; 4&#44; for problems in 2D&#44; and 8 for problems in 3D&#46;</p><p id="par0545" class="elsevierStylePara elsevierViewall">As for the right hand-sides of <a class="elsevierStyleCrossRef" href="#eq0135">Eqs&#46; &#40;4&#46;14&#41;</a>&#44; all they can be obtained by successively applying to f&#95;&#59;&#916; some of the operators that have already been discussed&#46; Recalling <a class="elsevierStyleCrossRef" href="#eq0135">Eq&#46; &#40;4&#46;14&#41;</a>&#44; we have<elsevierMultimedia ident="eq0330"></elsevierMultimedia></p><p id="par0550" class="elsevierStylePara elsevierViewall">The computation of Rf&#8994;&#95;&#59; does not present any difficulty and the evaluation of the actions of A&#95;&#95;&#928;&#928;&#8722;1 and A&#95;&#95;&#916;&#928; were already analyzed&#46;</p></span></span><span id="sec0085" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">7</span><span class="elsevierStyleSectionTitle" id="sect0105">Numerical Results</span><p id="par0555" class="elsevierStylePara elsevierViewall">Taking into account the general description of the DVS-framework given of <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>&#44; it can be seen that each one of the DVS-algorithms is uniquely defined by&#58;<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0040"><span class="elsevierStyleLabel">1&#46;</span><p id="par0560" class="elsevierStylePara elsevierViewall">The original-matrix&#59;</p></li><li class="elsevierStyleListItem" id="lsti0045"><span class="elsevierStyleLabel">2&#46;</span><p id="par0565" class="elsevierStylePara elsevierViewall">The partition of the set of original-nodes&#44; which is induced by the <span class="elsevierStyleItalic">coarse-mesh</span> that is applied&#59; and</p></li><li class="elsevierStyleListItem" id="lsti0050"><span class="elsevierStyleLabel">3&#46;</span><p id="par0570" class="elsevierStylePara elsevierViewall">The set of constraints&#46;</p></li></ul></p><p id="par0575" class="elsevierStylePara elsevierViewall">In turn&#44; the original-matrix is determined by the partial differential equation&#44; or system of such equations&#44; the discretization method chosen and the <span class="elsevierStyleItalic">fine-mesh</span> adopted&#46; As explained in <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>&#44; the partition of the set of original-nodes depends when the <span class="elsevierStyleItalic">fine-mesh</span> has already been defined&#44; on the <span class="elsevierStyleItalic">coarse-mesh</span> &#40;i&#46;e&#46;&#44; the domain decomposition&#41; used&#46; The <span class="elsevierStyleItalic">coarse-mesh</span> is constituted by a family of non-overlapping subdomains &#123;&#937;<span class="elsevierStyleInf">1</span>&#44;&#46;&#46;&#46;&#44; &#937;<span class="elsevierStyleItalic"><span class="elsevierStyleInf">E</span></span>&#125; of &#937;&#44; the domain of definition of the boundary-value problem to be solved&#46; In all the examples that are presented in this article&#44; the <span class="elsevierStyleItalic">constraints</span> are fully determined by the <span class="elsevierStyleItalic">primal-nodes</span> and consist in requiring continuity of the <span class="elsevierStyleItalic">derived-vectors</span> at them&#46;</p><p id="par0580" class="elsevierStylePara elsevierViewall">Several codes were developed to treat the examples&#44; which were written in C&#43;&#43; language&#44; using the MPI library for the communications&#46; In the computational implementations&#44; the methods of solution used to treat the <span class="elsevierStyleItalic">original-problems</span> are&#58; CGM&#44; when such a linear system is symmetric and positive-definite and GMRES when the discrete system is non-symmetric or indefinite&#46; Both are applied with a tolerance of 10<span class="elsevierStyleSup">&#8722;6</span>&#46; Each <span class="elsevierStyleItalic">DVS-algorithm</span> was applied to each one of the examples considered&#44; except for that referring to elasticity&#46;</p><p id="par0585" class="elsevierStylePara elsevierViewall">The results obtained for Examples 1 to 5 are summarized in <a class="elsevierStyleCrossRef" href="#tbl0005">Tables 1</a> to <a class="elsevierStyleCrossRef" href="#tbl0025">5</a>&#44; respectively&#46; In them&#44; the acronym <span class="elsevierStyleItalic">dof</span> stands for to the number of degrees of freedom of the <span class="elsevierStyleItalic">original problem</span>&#44; but it should be mentioned that the procedures used to treat such examples are such that the nodes that lie on the external boundary do not contribute to the <span class="elsevierStyleItalic">dof</span>&#46; The notation to indicate the meshes that were adopted is as follows&#58; In 2D cases&#44; we use <span class="elsevierStyleItalic">&#40;nxm&#41;x&#40;qxr&#41;</span>&#44; where <span class="elsevierStyleItalic">&#40;nxm&#41;</span> refers to the <span class="elsevierStyleItalic">coarsemesh</span>&#44; while <span class="elsevierStyleItalic">&#40;qxr&#41;</span> to the <span class="elsevierStyleItalic">fine-mesh</span>&#59; and similarly&#44; in 3D cases&#44; we use <span class="elsevierStyleItalic">&#40;nxmxp&#41;x&#40;qxrxs&#41;</span>&#44; where <span class="elsevierStyleItalic">&#40;nxmxp&#41;</span> define the <span class="elsevierStyleItalic">coarse-mesh</span> and <span class="elsevierStyleItalic">&#40;qxrxs&#41;</span> the <span class="elsevierStyleItalic">fine-mesh</span>&#46; The constrains are imposed on the primal nodes&#44; in all of our experiments the primal nodes were located at vertex in 2D and at edges in 3D of the subdomains&#44; this coinciding with the algorithm &#8220;D&#8221; in &#91;<a class="elsevierStyleCrossRef" href="#bib0165">Toselli <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#93;&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><elsevierMultimedia ident="tbl0025"></elsevierMultimedia><p id="par0590" class="elsevierStylePara elsevierViewall">Each Table contains at most ten columns&#46; The first four indicate respectively&#58; 1&#41; the meshes used&#44; 2&#41; the number of subdomains of the <span class="elsevierStyleItalic">coarsemesh</span>&#44; 3&#41; the <span class="elsevierStyleItalic">dof</span>&#44; and 4&#41; the number of <span class="elsevierStyleItalic">primal-nodes</span> used&#46; The figures appearing in columns 5 to 9 correspond to the number of iterations that were required for convergence of each one of the algorithms applied&#46; Columns 9 and 10 were only included in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>&#46; For Example 3&#44; in order to cover a wide range of values of the Peclet-number&#44; the diffusion coefficient in <a class="elsevierStyleCrossRef" href="#eq0345">Eq&#46; &#40;7&#46;3&#41;</a>&#44; <span class="elsevierStyleItalic">&#957;</span>&#44; was varied and the tenth column in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a> indicates the different values of <span class="elsevierStyleItalic">&#957;</span> for which the corresponding boundary-value problem was solved&#46; Furthermore&#44; the results obtained when the DVS-algorithms were applied were compared with those obtained in &#91;<a class="elsevierStyleCrossRef" href="#bib0005">Da Concei&#231;&#227;o <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2006</a>&#93; for the same problem&#44; using the standard version of BDDC&#46;<a name="p303"></a></p><elsevierMultimedia ident="tbl0015"></elsevierMultimedia><span id="sec0090" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">7&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0110">Application of the DVS-algorithms to a Single-Equation</span><p id="par0595" class="elsevierStylePara elsevierViewall">The applicability of the <span class="elsevierStyleItalic">DVS-algorithms</span> is wide&#44; as previously said it can be applied to general equation systems&#46; In <a class="elsevierStyleCrossRef" href="#sec0015">Section 3</a>&#44; it was <span class="elsevierStyleItalic">n</span> annou<span class="elsevierStyleItalic">n</span>ced that in this paper we present examples for which <span class="elsevierStyleItalic">d</span>&#44; the number of equation s of th e system&#44; is one and three&#46; In this Subsection the examples for which <span class="elsevierStyleItalic">d</span>&#61;1 will be discussed&#44; leaving for the next Subsection the treatment of static-elasticity models&#44; for which <span class="elsevierStyleItalic">d</span>&#61;3&#46;</p><p id="par0600" class="elsevierStylePara elsevierViewall">Four boundary value problems corresponding to a single-equation will be presented&#46; The first two are symmetric and positive definite boundaryvalue problems&#44; whose definition involves the Laplace differential operator&#46; The other two correspond to advection-diffusion transport&#44; and the corresponding boundary-value problems are non-symmetric and indefinite&#46; The discretization methods used in this Subsection are based on central finite differences &#40;CFD&#41;&#44; which are directly applicable to the symmetric problems&#46; To apply CFD to the advection-diffusion problems it was necessary to stabilize the advection-diffusion differential-operator and to this end artificial diffusion was incorporated&#46;</p><p id="par0605" class="elsevierStylePara elsevierViewall">Despite the simplicity of the examples presented in this Subsection&#44; they are very important because a wide range of geophysical systems give rise to similar problems &#91;<a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Pinder&#44; 2012</a>&#93;&#46; The diversity of physical interpretations of the boundary-value problems here discussed is enormous&#46; All the differential operators involved can be classified as advection-diffusion operators&#44; since Laplace operator is obtained from the general advection-diffusion differential-operator when the transport-velocity vanishes&#46; Transport processes of heat and solutes occur in a great diversity of geophysical systems&#46; However&#44; the physical processes governed by such differential-equations go far beyond transport phenomena&#46;</p><p id="par0610" class="elsevierStylePara elsevierViewall">Example 1&#46; Poisson equation in two-dimensions&#46;<elsevierMultimedia ident="eq0335"></elsevierMultimedia></p><p id="par0615" class="elsevierStylePara elsevierViewall">We can see from <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#44; that the four algorithms perform very well as the number of subdomains and the degrees of freedom &#40;dof&#41; are increased&#46; In this example&#44; the DVS-DUAL algorithm presents the best performance&#44; requiring only 11 iterations from 12x12 until 30x30 subdomains&#44; and the same number of dof&#46; All other algorithms show similar behavior&#46; The numerical solution of this example can be seen in the <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>&#46;</p><elsevierMultimedia ident="fig0025"></elsevierMultimedia><p id="par0620" class="elsevierStylePara elsevierViewall">Example 2&#46; Similar to Example 1&#44; but it is formulated in a 3D domain&#46;<elsevierMultimedia ident="eq0340"></elsevierMultimedia></p><p id="par0625" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>&#44; we observe a similar performance of the algorithms as in the two-dimensional case&#46; One more time the DVS-DUAL algorithm presents a little better behavior with respect all others&#46;</p><elsevierMultimedia ident="tbl0010"></elsevierMultimedia><p id="par0630" class="elsevierStylePara elsevierViewall">Example 3&#46; The boundary-value problem treated is&#58;<elsevierMultimedia ident="eq0345"></elsevierMultimedia></p><p id="par0635" class="elsevierStylePara elsevierViewall">This is an advection-diffusion transport problem in 2D&#44; for which the differential operator is not self-adjoint&#46;</p><p id="par0640" class="elsevierStylePara elsevierViewall">This example is very interesting because it contains diffusion and advection terms&#44; which are common in several complex geophysics phenomena&#46; In this example&#44; the P&#233;clet number is defined as Pe&#61;b&#95;&#59;L&#47;&#957; where L is a characteristic length &#40;in this case L &#61; 1&#41;&#46; We also define a local P&#233;clet number as Peh&#61;b&#95;&#59;h&#47;&#957;&#46; Using these definitions&#44; fixing the global partition to h&#61;1&#47;512&#44; and the varying the viscosity from 0&#46;01 to 0&#46;0001&#44; we have that the P&#233;clet number varies from 316 to 316&#44;227&#44; and the local P&#233;clet number varies from 0&#46;617 to 617&#46; In this case the linear system is non-symmetric&#44; therefore we choose the GMRES method with a tolerance of 10<span class="elsevierStyleSup">&#8722;6</span>&#46;</p><p id="par0645" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a> presents the results that the <span class="elsevierStyleItalic">DVS-algorithms</span> yielded and compares them with those obtained in &#91;<a class="elsevierStyleCrossRef" href="#bib0005">Da Concei&#231;&#227;o <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2006</a>&#93;&#46; We observe that&#44; with fixed <span class="elsevierStyleItalic">coarse</span> and <span class="elsevierStyleItalic">fine meshes</span>&#44; as the viscosity coefficient is reduced&#44; so that the P&#233;clet number increases&#44; generally the iterations required for convergence reduce&#46; Increasing the P&#233;clet number implies that the effect of the advection term enlarges&#44; and the numerical solution generally becomes unstable&#46; However&#44; the performance of the discretization strategy based on CFD combined with stabilization of the numerical-scheme by means of artificial viscosity is resilient to P&#233;clet-number variations&#46; For comparison purposes&#44; the examples presented here were chosen to be the same as those<a name="p304"></a> presented in &#91;<a class="elsevierStyleCrossRef" href="#bib0005">Da Concei&#231;&#227;o <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2006</a>&#93;&#44; where the standard BDDC algorithm was applied with the same set of constraints&#59; namely&#44; the same set of subdomains and vertex nodes were chosen to be <span class="elsevierStyleItalic">primal</span>&#46; As can be seen in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>&#44; when the comparison criterion is based on the number of iterations required for convergence&#44; the observed performance of the <span class="elsevierStyleItalic">DVS-algorithms</span> in these examples is slightly better than that of the standard BDDC algorithm&#46; Finally&#44; an illustration of the kind of numerical solution obtained is shown in <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>&#46;<a name="p305"></a><elsevierMultimedia ident="fig0030"></elsevierMultimedia></p><elsevierMultimedia ident="fig0035"></elsevierMultimedia><p id="par0650" class="elsevierStylePara elsevierViewall">The relative-residual decay for a coarse mesh &#40;16&#215;16&#41; and several fine meshes is presented in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>&#46; We consider in these computations b &#61; &#40;1&#44; 3&#41; and <span class="elsevierStyleItalic">&#957;</span> &#61; 0&#46;00001&#44; in such a way that P<span class="elsevierStyleItalic">e</span> &#61; 3&#46;16<span class="elsevierStyleItalic">e</span>&#43;5&#46; We observe that the best convergence is obtained when the fine mesh is increased&#44; and the convergence slows when the dof occurring in the subdomains is reduced&#46;</p><elsevierMultimedia ident="fig0040"></elsevierMultimedia><p id="par0655" class="elsevierStylePara elsevierViewall">Example 4&#46; The boundary-value problem treated is&#58;<elsevierMultimedia ident="eq0350"></elsevierMultimedia></p><p id="par0660" class="elsevierStylePara elsevierViewall">This is an advection-diffusion transport problem in 3D&#44; for which the differential operator is not self-adjoint&#46;</p><p id="par0665" class="elsevierStylePara elsevierViewall">The diffusion and advection-diffusion differential-operator appears in the equations of the examples presented above&#46; They are very important in natural and industrial phenomena&#46; For example&#44; the flow and transport of solutes in subsurface groundwater&#44; the movement of aerosol and trace gases in the atmosphere&#44; mixing of fluids in processes of crystal growth&#44; among many other important applications &#91;<a class="elsevierStyleCrossRef" href="#bib0160">Tood&#44; 1980</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0150">Pinder <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2006</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0095">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1969</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0100">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1973</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0105">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 1977</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0110">Herrera G&#46;S&#46; <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0020">L&#46;M&#46; de la Cruz <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2006</a>&#93;&#46; In all our examples&#44; we have shown that the <span class="elsevierStyleItalic">DVS-algorithms</span> obtain the numerical solution efficiently on parallel machines&#46; In this respect&#44; we remark that for advection-diffusion problems the matrices of the discrete linear systems are non-symmetric&#46;</p></span><span id="sec0095" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">7&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0115">Application to a System-Equations</span><p id="par0670" class="elsevierStylePara elsevierViewall">We use the <span class="elsevierStyleItalic">DVS-framework</span> to solve a Dirichlet boundary value problem&#44; where displacements are zero over the boundary of the elastic body that occupies the domain <span class="elsevierStyleItalic">&#937;</span> of the physical space&#46;<a name="p306"></a> Over each one of such subdomains is solved a local problem by FEM&#44; using linear functions as basis&#46; On each node <span class="elsevierStyleItalic">&#945;</span> of the mesh is defined a vector valued function u&#95;&#59;&#945; with each component identified as <span class="elsevierStyleItalic">u<span class="elsevierStyleInf">&#945;i</span></span> for <span class="elsevierStyleItalic">i</span>&#61;1&#44; 2&#44; 3&#46;<elsevierMultimedia ident="tbl0020"></elsevierMultimedia></p><p id="par0675" class="elsevierStylePara elsevierViewall">Because our operators are symmetric and positive definite&#44; we use CGM as an iterative procedure to solve those linear systems of equations that we have defined in the DVS framework&#46;</p><p id="par0680" class="elsevierStylePara elsevierViewall">The code used in the previous section&#44; which was originally developed to solve a single equation using finite differences&#44; was adapted for solving systems of equations with FEM&#46; We added the corresponding functionality in order to be able to solve systems of equations&#44; in this case the elasticity problem&#46;</p><p id="par0685" class="elsevierStylePara elsevierViewall">Example 5&#46; A system of partial differential equations in three-dimensions has also been treated&#46; This is the system of differential equations of static elasticity&#59; namely&#58;<a name="p307"></a><elsevierMultimedia ident="eq0355"></elsevierMultimedia></p><p id="par0695" class="elsevierStylePara elsevierViewall">which was subject to the following Dirichlet boundary conditions&#58;<elsevierMultimedia ident="eq0360"></elsevierMultimedia></p><p id="par0700" class="elsevierStylePara elsevierViewall">The domain of study for our numerical experiments is a homogeneous isotropic linearly elastic unitary cube&#46; In all of our experiments the primal nodes were located at edges of the subdomains&#44; which is enough for A&#95;&#95;t not being singular&#46;</p><p id="par0705" class="elsevierStylePara elsevierViewall">We consider constant coefficients <span class="elsevierStyleItalic">&#955;</span> and <span class="elsevierStyleItalic">&#956;</span> equal to one&#46; With these conditions we have a problem that has analytical solution&#44; and is written as follows&#58;<elsevierMultimedia ident="eq0365"></elsevierMultimedia></p><p id="par0710" class="elsevierStylePara elsevierViewall">The <a class="elsevierStyleCrossRef" href="#tbl0025">Tables 5</a>&#44; summarizes the numerical results obtained using the DVS methods with a tolerance of 10<span class="elsevierStyleSup">-7</span>&#46;</p></span></span><span id="sec0100" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">8</span><span class="elsevierStyleSectionTitle" id="sect0120">Conclusions</span><p id="par0715" class="elsevierStylePara elsevierViewall">Mathematical models of many geophysical systems lead to a great variety of partial differential equations &#40;PDEs&#41; whose solution methods are based on the computational processing of largescale algebraic systems &#91;<a class="elsevierStyleCrossRef" href="#bib0050">Herrera and Pinder&#44; 2012</a>&#93;&#46; Parallel computing is outstanding among the new computational tools and&#44; in order to effectively use the most advanced computers available today&#44; massively parallel software is required&#46; Domain decomposition methods &#40;DDMs&#41; have been developed precisely for effectively treating PDEs in parallel &#91;<a class="elsevierStyleCrossRef" href="#bib0010">DDM Organization&#44; 2012</a>&#93;&#46; What domain decomposition methods ideally intend to do has been summarized in this paper in the <span class="elsevierStyleItalic">&#8220;DDM-paradigm&#8221;</span>&#58; to develop algorithms that &#8216;<span class="elsevierStyleItalic">obtain the <span class="elsevierStyleUnderline">global</span> solution by exclusively solving <span class="elsevierStyleUnderline">local</span> problems</span>&#8217;&#46;</p><p id="par0720" class="elsevierStylePara elsevierViewall">In conclusion&#44; in this paper&#58;<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0055"><span class="elsevierStyleLabel">1&#46;</span><p id="par0725" class="elsevierStylePara elsevierViewall">We have presented a <span class="elsevierStyleItalic">non-overlapping discretization</span> method &#40;the <span class="elsevierStyleItalic">DVS-discretization</span>&#41; -in the sense that it uses a system of nodes such that each one of them belongs to one and only one subdomain of the <span class="elsevierStyleItalic">coarse-mesh</span>- applicable to a wide class of well-posed boundary problems associated with elliptic systems of equations&#46; In particular&#44; the differential operators may be symmetric&#44; non-symmetric or indefinite &#40;nonpositive-definite&#41;&#59;</p></li><li class="elsevierStyleListItem" id="lsti0060"><span class="elsevierStyleLabel">2&#46;</span><p id="par0730" class="elsevierStylePara elsevierViewall">Four algorithms &#8211;the <span class="elsevierStyleItalic">DVS-algorithms</span> &#91;<a class="elsevierStyleCrossRef" href="#bib0060">Herrera <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2011</a>&#93;<span class="elsevierStyleItalic">-</span>&#44; which were derived using the <span class="elsevierStyleItalic">DVS-discretization</span> and achieve the <span class="elsevierStyleItalic">DDM-paradigm</span> have been explained&#46; Two of them are the result of using the BDDC and FETI-DP algorithms after applying <span class="elsevierStyleItalic">DVS-discretization</span> to the boundary value problem considered&#46; The other two are obtained when two new algorithms&#44; which had not been reported previously in the literature&#44; were used instead&#59;</p></li><li class="elsevierStyleListItem" id="lsti0065"><span class="elsevierStyleLabel">3&#46;</span><p id="par0735" class="elsevierStylePara elsevierViewall">Numerical procedures that permit achieving the <span class="elsevierStyleItalic">DDM-paradigm</span> with each one of the <span class="elsevierStyleItalic">DVS-algorithms</span> have been also presented&#59;</p></li><li class="elsevierStyleListItem" id="lsti0070"><span class="elsevierStyleLabel">4&#46;</span><p id="par0740" class="elsevierStylePara elsevierViewall">Codes were developed and applied to several boundary values problems that occur in the modeling of certain geophysical phenomena&#44; such as transport of solutes by both&#44; free-fluids and fluids in a porous medium&#46; We also present results for a static elasticity problem&#44; which thereby illustrates the application of the algorithms to systems of differential equations&#59; and</p></li><li class="elsevierStyleListItem" id="lsti0075"><span class="elsevierStyleLabel">5&#46;</span><p id="par0745" class="elsevierStylePara elsevierViewall">Besides their attractive parallelization properties&#44; in the numerical examples the <span class="elsevierStyleItalic">DVS-algorithms</span> exhibited significantly improved numerical performance with respect to standard versions of BDDC and FETI-DP&#46;</p></li></ul></p></span></span>"
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          "titulo" => "DVS Framework&#58; A Summary"
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          "titulo" => "The Derived-Vector Space &#40;DVS&#41;"
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          "titulo" => "The Transformed Problem"
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          "titulo" => "The DVS-Algorithms"
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                  "titulo" => "The DVS Version of BDDC"
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          "titulo" => "Numerical Procedures Fulfilling the DDM-Paradigm"
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              "titulo" => "Comment on the DVS Numerical Procedures"
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              "titulo" => "Application of S&#95;&#95;&#8722;1"
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              "titulo" => "Application of the DVS-algorithms to a Single-Equation"
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              "titulo" => "Application to a System-Equations"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Los modelos matem&#225;ticos de muchos sistemas geof&#237;sicos requieren el procesamiento de sistemas algebraicos de gran escala&#46; Las herramientas computacionales m&#225;s avanzadas est&#225;n masivamente paralelizadas&#46; El software m&#225;s efectivo para resolver ecuaciones diferenciales parciales en paralelo intenta alcanzar el <span class="elsevierStyleItalic">paradigma de los m&#233;todos de descomposici&#243;n de dominio</span>&#44; que hasta ahora se hab&#237;a mantenido como un anhelo no alcanzado&#46; Sin embargo&#44; un grupo de cuatro algoritmos &#8211;los <span class="elsevierStyleItalic">algoritmos DVS</span>- que lo alcanzan y que tiene aplicabilidad muy general se ha desarrollado recientemente&#46; Este art&#237;culo est&#225; dedicado a presentarlos y a ilustrar su aplicaci&#243;n a problemas que se presentan frecuentemente en la investigaci&#243;n y el estudio de la Geof&#237;sica&#46;</p></span>"
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        "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Mathematical models of many geophysical systems are based on the computational processing of large-scale algebraic systems&#46; The most advanced computational tools are based on massively parallel processors&#46; The most effective software for solving partial differential equations in parallel intends to achieve the <span class="elsevierStyleItalic">DDM-paradigm</span>&#46; A set of four algorithms&#44; the <span class="elsevierStyleItalic">DVS-algorithms</span>&#44; which achieve it&#44; and of very general applicability&#44; has recently been developed and here they are explained&#46; Also&#44; their application to problems that frequently occur in Geophysics is illustrated&#46;</p></span>"
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                  <table border="0" frame="\n
                  \t\t\t\t\tvoid\n
                  \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head  " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">SUBDOMAINS&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DOF&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">PRIMALS&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-BDDC&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-FETI-DP&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-PRIMAL&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">DVS-DUAL&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="middle">&#40;2&#215;2&#41; &#215; &#40;2&#215;2&#41;&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">4&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">9&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">1&nbsp;\t\t\t\t\t\t\n
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                  """
              ]
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                0 => "xTab795119.png"
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        "descripcion" => array:1 [
          "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Number of iterations made by the four DVS algorithms&#46; The primal nodes were located at the vertices of subdomains&#46;</p>"
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      ]
      8 => array:7 [
        "identificador" => "tbl0010"
        "etiqueta" => "Table 2"
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        "mostrarFloat" => true
        "mostrarDisplay" => false
        "tabla" => array:1 [
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                  \t\t\t\t\tvoid\n
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                  """
              ]
              "imagenFichero" => array:1 [
                0 => "xTab795116.png"
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          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0045" class="elsevierStyleSimplePara elsevierViewall">Number of iterations made by the four DVS algorithms&#46; The primal nodes were located at edge&#46;</p>"
        ]
      ]
      9 => array:7 [
        "identificador" => "tbl0015"
        "etiqueta" => "Table 3"
        "tipo" => "MULTIMEDIATABLA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "tabla" => array:1 [
          "tablatextoimagen" => array:1 [
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                0 => """
                  <table border="0" frame="\n
                  \t\t\t\t\tvoid\n
                  \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head  " align="left" valign="middle" scope="col" style="border-bottom: 2px solid black">PARTITION&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">9&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">8&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="middle">&#40;7&#215;7&#215;7&#41; &#215; &#40;7&#215;7&#215;7&#41;&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">12&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">8&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">10&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">8&nbsp;\t\t\t\t\t\t\n
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                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">8&nbsp;\t\t\t\t\t\t\n
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  "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714788?idApp=UINPBA00004N"
]

Article information

ISSN: 00167169
Original language: English

DOI:10.1016/S0016-7169(13)71478-8

The statistics are updated each day

Year/Month	Html	Pdf	Total

2024 November	2	2	4
2024 October	7	8	15
2024 September	9	15	24
2024 August	9	13	22
2024 July	8	4	12
2024 June	12	6	18
2024 May	10	4	14
2024 April	42	13	55
2024 March	18	12	30
2024 February	12	6	18
2024 January	60	4	64
2023 December	18	7	25
2023 November	16	7	23
2023 October	9	8	17
2023 September	5	1	6
2023 August	10	4	14
2023 July	10	7	17
2023 June	7	2	9
2023 May	4	10	14
2023 April	6	5	11
2023 March	5	9	14
2023 February	1	2	3
2023 January	7	15	22
2022 December	10	6	16
2022 November	7	13	20
2022 October	9	12	21
2022 September	12	8	20
2022 August	18	9	27
2022 July	10	12	22
2022 June	11	11	22
2022 May	13	19	32
2022 April	10	12	22
2022 March	4	15	19
2022 February	6	4	10
2022 January	18	4	22
2021 December	3	9	12
2021 November	10	10	20
2021 October	8	9	17
2021 September	4	11	15
2021 August	5	6	11
2021 July	7	9	16
2021 June	6	9	15
2021 May	8	4	12
2021 April	23	11	34
2021 March	16	3	19
2021 February	3	5	8
2021 January	12	10	22
2020 December	8	7	15
2020 November	5	7	12
2020 October	6	9	15
2020 September	6	3	9
2020 August	13	3	16
2020 July	16	1	17
2020 June	3	2	5
2020 May	7	2	9
2020 April	5	3	8
2020 March	3	3	6
2020 February	4	2	6
2020 January	2	4	6
2019 December	9	5	14
2019 November	0	1	1
2019 October	3	2	5
2019 September	4	4	8
2019 August	3	3	6
2019 July	2	11	13
2019 June	8	22	30
2019 May	41	61	102
2019 April	20	22	42
2019 March	4	2	6
2019 February	0	6	6
2019 January	3	2	5
2018 December	1	10	11
2018 November	4	3	7
2018 October	3	1	4
2018 September	15	2	17
2018 August	10	0	10
2018 July	1	2	3
2018 June	4	0	4
2018 May	4	6	10
2018 April	1	0	1
2018 March	0	1	1
2018 February	1	0	1
2018 January	0	2	2
2017 December	1	2	3
2017 November	1	1	2
2017 October	4	0	4
2017 September	3	1	4
2017 August	5	10	15
2017 July	4	2	6
2017 June	12	4	16
2017 May	17	0	17
2017 April	11	5	16
2017 March	7	61	68
2017 February	6	3	9
2017 January	6	0	6
2016 December	5	2	7
2016 November	6	1	7
2016 October	1	4	5
2016 September	1	2	3
2016 August	2	0	2
2016 July	3	0	3
2016 June	4	1	5
2016 May	0	10	10
2016 April	0	3	3
2016 March	5	10	15
2016 February	5	8	13
2016 January	1	14	15
2015 December	3	7	10
2015 November	2	9	11
2015 October	11	11	22
2015 September	9	4	13
2015 August	1	1	2
2015 July	9	2	11
2015 June	0	3	3
2015 May	1	2	3
2015 April	3	6	9
2015 March	1	0	1

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