A Discontinuous Galerkin like Coarse Space correction for Domain Decomposition Methods with continuous local spaces: the DCS-DGLC Algorithm

Abstract

Abstract. In this paper, we are interested in scalable Domain Decomposition Methods (DDM). To this end, we introduce and study a new Coarse Space Correction algorithm for Optimized Schwarz Methods(OSM): the DCS-DGLC algorithm. The main idea is to use a Discontinuous Galerkin like formulation to compute a discontinuous coarse space correction. While the local spaces remain continuous, the coarse space should be discontinuous to compensate the discontinuities introduced by the OSM at the interface between neighboring subdomains. The discontinuous coarse correction algorithm can be used not only with OSM but also with any one-level DDM that produce discontinuous iterates. While ideas from Discontinuous Galerkin(DG) are used in the computation of the coarse correction, the final aim of the DCS-DGLC algorithm is to compute in parallel the discrete solution to the classical non-DG finite element problem. Résumé. Dans cet article, nous nous intéressons aux Méthods de Décomposition de Domaines (DDM)scalables.À cette fin, nous introduisons etétudions un nouvel algorithme, le DCS-DGLC, de correction grossière pour les méthodes de Schwarz optimisées. L'idée principale est d'utiliser une formulation apparentée aux méthodes de Galerkin discontinues pour calculer une correction grossière discontinue. Alors même que les espaces locaux restent continus, l'espace grossier est choisi discontinu afin de pouvoir compenser les discontinuités introduites par les OSM aux interfaces entre sous-domaines voisins. Cet algorithme de correction grossière discontinue peutêtre employé non seulement avec les OSM mais aussi avec toute DDM de un niveau qui produit des itérées discontinues. Bien que l'algorithme s'inspire des méthodes de Galerkin discontinues, le but final de l'agorithme est de calculer en parallèle la solution discreteà la formulationéléments finis classique, sans DG.

A Discontinuous Coarse Space (DCS) Algorithm for Cell Centered Finite Volume Based Domain Decomposition Methods: The DCS-RJMin Algorithm

“…In that proceeding, one possible algorithm, the DCS-DMNV (Discontinuous Coarse Space Dirichlet Minimizer Neumann Variational), was described at the continuous level and at the discrete level for Finite Element Methods on a non-overlapping Domain Decomposition. In [17], a similar method, the DCS-DGLC algorithm was proposed. Both the DCS-DMNV and the DCS-DGLC are well suited to finite element discretizations.…”