Convergence Analysis of GMRES for the SUPG Discretized Convection-Diffusion Model Problem

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Author(s) : Jörg Liesen , Zdenek Strakos

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 26-2003

MSC 2000

65F10 Iterative methods for linear systems
65F15 Eigenvalues, eigenvectors
65N22 Solution of discretized equations
65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Abstract :
When GMRES is applied to streamline-diffusion 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 norms. We concentrate on a well-known model problem with a constant velocity field parallel to one of the axes and with Dirichlet boundary conditions. Instead of the eigendecomposition of the system matrix we use the simultaneous diagonalization of the matrix blocks to 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.

Keywords : convection-diffusion problem, SUPG discretization, GMRES, rate of convergence, ill conditioned eigenvectors, nonnormality, tridiagonal Toeplitz matrices