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Iterative solvers for generalized finite element solution of boundary‐value problems
Author(s) -
Mohamed M Shadi,
Seaid Mohammed,
Bouhamidi Abderrahman
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2205
Subject(s) - finite element method , mathematics , iterative method , boundary value problem , linear system , mathematical optimization , generalized minimal residual method , matrix free methods , sparse matrix , mathematical analysis , physics , thermodynamics , quantum mechanics , gaussian
Summary Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill‐conditioned linear systems that are associated with these methods. In this study, we investigate the performance of a class of iterative solvers for the generalized finite element solution of time‐dependent boundary‐value problems. A fully implicit time‐stepping scheme is used for the time integration in the finite element framework. As enrichment, we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers, we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared with dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix–vector products are involved in their implementation. Two test examples for boundary‐value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill‐conditioned linear systems resulting from the generalized finite element methods for time domain problems.