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