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• W2402503558 abstract "In this paper, we present two variants of the additive Schwarz method for a Crouzeix-Raviart finite volume element (CRFVE) discretization of second-order elliptic problems with discontinuous coefficients, where the discontinuities may be across subdomain boundaries. The preconditioner in one variant is symmetric, while in the other variant it is nonsymmetric. The proposed methods are quasi optimal, in the sense that the convergence of the preconditioned GMRES iteration in both cases depend only poly-logarithmically on the ratio of the subdomain size to the mesh size. 1. Introduction. We introduce and analyze two variants of the additive Schwarz method (ASM) for a Crouzeix-Raviart finite volume element (CRFVE) discretization of second-order elliptic partial differential equations with discontinuous coefficients, where the discontinuities may be across subdomain boundaries. Problems of this type play an important role in scientific computing. Discontinuities or jumps in the coefficient cause the performance of any standard iterative method to deteriorate as the jump increases. The resulting system, which is in general nonsymmetric, is solved using the preconditioned generalized minimal residual (or preconditioned GMRES) method. We consider two variants of the ASM preconditioner, i.e., a symmetric and a nonsymmetric variant. The proposed methods are almost optimal in the sense that the convergence of the GMRES iterations in both cases depends only poly-logarithmically on the ratio of the subdomain size to the mesh size. The finite volume method divides the computational domain into a set of control volumes whose centroids typically correspond to the nodal points of a finite difference or a finite element discretization. Unlike the finite difference and the finite element method, the solution from a finite volume discretization ensures conservation of certain quantities such as mass, momentum, energy and species. This property is satisfied exactly for each control volume in the domain as well as the whole of the computational domain, connecting the solution to the physics of the system, and, as a consequence, making the method more attractive. There are two types of finite volume methods: one which is based on the finite difference discretization (also known as the finite volume method), and one which is based on the finite element discretization (also known as the finite volume element (FVE) method). In the later case the approximation of the solution is sought in a finite element space, and therefore it can be considered as a Petrov-Galerkin finite element method. Typically, the finite element space is defined on a mesh, called the primal mesh, and the equations are discretized on a mesh which is dual to the primal mesh. The CRVFE method is a variant of the FVE method, where the solution is sought in the Crouzeix-Raviart (CR) or the P1 nonconforming finite element spaces, as opposed to, say, the" @default.
• W2402503558 created "2016-06-24" @default.
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• W2402503558 date "2015-01-01" @default.
• W2402503558 modified "2023-09-27" @default.
• W2402503558 title "EDGE-BASED SCHWARZ METHODS FOR THE CROUZEIX-RAVIART FINITE VOLUME ELEMENT DISCRETIZATION OF ELLIPTIC PROBLEMS ∗" @default.
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