Nonoverlapping discretization methods for partial differential equations

Herrera, Ismael; Cruz, Luis M. de la; Rosas-Medina, Alberto

doi:10.1002/num.21852

Cited by 4 publications

(25 citation statements)

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“…The theoretical DVS-framework is very elegant; in it, the algebraic operations can be carried out systematically and with great simplicity. Furthermore, many simplifying algebraic results have been obtained in previous work (Herrera et al, 2014 and. To optimize the communications and processing time a purely algebraic criticalroute is defined, which profits much from such algebraic results previously obtained.…”

Section: Overview Of Dvs-softwarementioning

confidence: 99%

“…The emergence of parallel computing prompted on the part of the computational-modeling community a continued and systematic effort with the purpose of harnessing it for the endeavor of solving the mathematical models of scientific and engineering systems. Very early after such an effort began, it was recognized that domain decomposition methods (DDM) were the most effective technique for applying parallel computing to the solution of partial differential equations (DDM Organization 1988-2014, since such an approach drastically simplifies the coordination of the many processors that carry out the different tasks and also reduces very much the requirements of information-transmission between them (Toselli and Widlund, 2005) (Farhat et al, 2000).…”

Section: Introductionmentioning

confidence: 99%

“…When a DDM is applied, firstly a discretization of the mathematical model is carried out in a fine-mesh and, afterwards, a coarse-mesh is introduced, which properly constitutes the domain-decomposition. The 'DDM-paradigm', a paradigm for domain decomposition methods concomitant with the paradigm of parallel-computing software (Herrera et al, 2014), consists in 'obtaining the global solution by solving local problems exclusively' (a local problem is one defined separately in a subdomain of the coarse-mesh). Stated in a simplistic manner, the basic idea is that, when the DDM-paradigm is satisfied, full parallelization can be achieved by assigning each subdomain to a different processor.…”

Section: Introductionmentioning

confidence: 99%

“…However, even in this kind of methods different subdomains are linked by interface nodes that are shared by several subdomains and, therefore, nonoverlapping DDMs are actually overlapping when seen from the perspective of the nodes used in the discretization. So, a more thorough uncoupling of the local problems and significant computational advantages should be expected if it were possible to carry out the discretization of the differential equations in a 'non-overlapping system of nodes' (Herrera et al, 2014); i.e., a set of nodes with the property that each one of them belongs to one and only one subdomain of the coarse-mesh (this is the mesh that constitutes a domain decomposition). In (Herrera et al, 2014), as in what follows, discretization methods that fulfill these conditions are referred to as nonoverlapping discretizations.…”

Section: Introductionmentioning

confidence: 99%

“…Such properties, as well as preliminary analysis of the algorithms, indicate that they should be extremely adequate to program the treatment of partial differential equations occurring in science and engineering models by the highly parallelized hardware of today. Although the DVS-algorithms have already been significantly developed and some examples have been previously treated (Herrera et al, 2014 andCarrillo-Ledesma et al, 2013), up to now no software that took full advantage of the DVS-algorithms had been developed. Clearly, to profit fully from such advances it is essential to develop software, carefully coded, which permit applying effectively the DVS-algorithms to problems of interest in science and engineering.…”

Section: Introductionmentioning

confidence: 99%

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An Innovative Tool for Effectively Applying Highly Parallelized Hardware To Problems of Elasticity

Herrera

Contreras

2016

GeofInt

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At present, the most efficient domain decomposition methods (DDM) are non-overlapping methods. The improved efficiency of such methods is due to the significant independence achieved by local problems when the subdomains are non-overlapping. However, standard discretizations applied up to now in non-overlapping DDMs use systems of nodes in which some of the nodes are shared by more than one subdomain of the domain decomposition. This is a limiting feature of the present state-of-the-art in these techniques and apparently further increases of the independence of local problems should be expected if this limiting characteristic was eliminated. In previous work, I. Herrera and co-workers have developed a new approach to domain decomposition methods: the ‘DVS framework’ that addresses this problem introducing a new discretization method, the ‘non-overlapping discretization method’, in which a non-overlapping system of nodes is used in the discrete formulation of the problem. Although the DVS algorithms have already been developed significantly, to profit from such advances it is essential to have available effective codes that permit their efficient implementation. As a further contribution in this line of research, in this paper we present and test software of such characteristics. The results here reported indicate that the DVS algorithms are very suitable for developing software that permits to apply effectively the most advanced hardware in parallel available at present to the solution of partial differential equations. Although the software here reported specifically treats static elasticity only, the DVS-algorithms are very eclectic and can be applied to a great diversity of problems after they have been discretized. Additional research work is being carried out oriented to develop general purpose codes based on the DVS algorithms.

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Section: Overview Of Dvs-softwarementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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