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15
A twolevel Schwarz preconditioner for heterogeneous problems
"... Coarse space correction is essential to achieve algorithmic scalability in domain decomposition methods. Our goal here is to build a robust coarse space for Schwarz–type preconditioners for elliptic problems with highly heterogeneous coefficients when the discontinuities are not just across but also ..."
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Coarse space correction is essential to achieve algorithmic scalability in domain decomposition methods. Our goal here is to build a robust coarse space for Schwarz–type preconditioners for elliptic problems with highly heterogeneous coefficients when the discontinuities are not just across but also along
A Discontinuous Galerkin like Coarse Space correction for Domain Decomposition Methods with continuous local spaces: the DCSDGLC Algorithm
, 2013
"... 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 DCSDGLC algorithm. The main idea is to use a Discontinuous Galerkin like formulation to compute a disc ..."
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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 DCSDGLC 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 not only be used with OSM but also be used with any onelevel 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 DCSDGLC algorithm is to compute in parallel the discrete solution to the classical nonDG finite element problem. Résumé Dans cet article, nous nous intéressons aux Méthods de Décomposition de Domaines (DDM) scalables. À cette fin, nous introduisons et étudions un nouvel algorithme, le DCSDGLC, 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 sousdomaines voisins. Cet algorithme de correction grossière discontinue peut être 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 discrete à la formulation éléments finis classique, sans DG.
Discontinuous Coarse Spaces for DDMethods with Discontinuous Iterates
"... Basic iterative domain decomposition methods (DDM) can only transmit information between direct neighbors. Such methods never converge in less iterations than the diameter of the connectivity graph between subdomains. Convergence rates are dependent on the number of subdomains, and thus algorithms a ..."
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Basic iterative domain decomposition methods (DDM) can only transmit information between direct neighbors. Such methods never converge in less iterations than the diameter of the connectivity graph between subdomains. Convergence rates are dependent on the number of subdomains, and thus algorithms are not scalable.
Nonlinear FETIDP and BDDC methods
 SIAM J. Sci. Comput
"... Abstract. New nonlinear FETIDP (dualprimal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new appro ..."
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Abstract. New nonlinear FETIDP (dualprimal finite element tearing and interconnecting) and BDDC (balancing domain decomposition by constraints) domain decomposition methods are introduced. In all these methods, in each iteration, local nonlinear problems are solved on the subdomains. The new approaches can significantly reduce communication and show a significantly improved performance, especially for problems with localized nonlinearities, compared to a standard Newton–Krylov–FETIDP or BDDC approach. Moreover, the coarse space of the nonlinear FETIDP methods can be used to accelerate the Newton convergence. It is also found that the new nonlinear FETIDP and nonlinear BDDC methods are not as closely related as in the linear context. Numerical results for the pLaplace operator are presented.
Additive average Schwarz method for the CrouzeixRaviart finite volume element discretization of elliptic problems
, 2013
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OPTIMIZED SCHWARZ AND 2LAGRANGE MULTIPLIER METHODS FOR MULTISCALE PDES
"... In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of the ..."
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In this article, we formulate and analyze a twolevel preconditioner for Optimized Schwarz and 2Lagrange Multiplier methods for PDEs with highly heterogeneous (multiscale) diffusion coefficients. The preconditioner is equipped with an automatic coarse space consisting of lowfrequency modes of the subdomain DirichlettoNeumann maps. Under a suitable change of basis, the preconditioner is a 2 × 2 block upper triangular matrix with the identity matrix in the upperleft block. We show that the spectrum of the preconditioned system is included in the disk having center z = 1/2 and radius r = 1/2 − , where 0 < < 1/2 is a parameter that we can choose. We further show that the GMRES algorithm applied to our heterogeneous system converges in O(1/) iterations (neglecting certain polylogarithmic terms). The number can be made arbitrarily large by automatically enriching the coarse space. Our theoretical results are confirmed by numerical experiments.
A Robust Two Level Domain Decomposition Preconditioner for Systems of PDEs
"... Received *****; accepted after revision +++++ Presented by Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial ..."
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Received *****; accepted after revision +++++ Presented by Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.
An Algebraic Local Generalized Eigenvalue in the Overlapping Zone Based Coarse Space: A first introduction
, 2011
"... Received *****; accepted after revision +++++ Presented by Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial ..."
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Received *****; accepted after revision +++++ Presented by Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.