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Main Title: GMRES convergence analysis for a convection-diffusion model problem
Author(s): Liesen, Jörg
Strakoš, Zdenek
Type: Article
Language Code: en
Abstract: When GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput.}, 7 (1986), pp. 856--869] is applied to streamline upwind Petrov--Galerkin (SUPG) discretized convection-diffusion problems, it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norm. Several approaches were made to understand this behavior. However, the existing analyses are solely based on the matrix of the discretized system and they do not take into account any influence of the right-hand side (determined by the boundary conditions and/or source term in the PDE). Therefore they cannot explain the length of the initial period of slow convergence which is right-hand side dependent. We concentrate on a frequently used model problem with Dirichlet boundary conditions and with a constant velocity field parallel to one of the axes. Instead of the eigendecomposition of the system matrix, which is ill conditioned, we use its orthogonal transformation into a block-diagonal matrix with nonsymmetric tridiagonal Toeplitz blocks and offer an explanation of GMRES convergence. We show how the initial period of slow convergence is related to the boundary conditions and address the question why the convergence in the second stage accelerates.
Issue Date: 2006
Date Available: 19-Dec-2017
DDC Class: 518 Numerische Analysis
Subject(s): convection-diffusion problem
streamline upwind Petrov--Galerkin discretization
rate of convergence
ill-conditioned eigenvectors
tridiagonal Toeplitz matrices
Journal Title: SIAM Journal on Scientific Computing
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa
Volume: 26
Issue: 6
Publisher DOI: 10.1137/S1064827503430746
Page Start: 1989
Page End: 2009
EISSN: 1095-7197
ISSN: 1064-8275
Appears in Collections:FG Numerische Lineare Algebra » Publications

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