Author(s) :
Jörg Liesen
,
Zdenek Strakos
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 262003
MSC 2000
 65F10 Iterative methods for linear systems

65F15 Eigenvalues, eigenvectors

65N22 Solution of discretized equations

65N30 Finite elements, RayleighRitz and Galerkin methods, finite methods
Abstract :
When GMRES is applied to streamlinediffusion upwind Petrov Galerkin
(SUPG) discretized convectiondiffusion problems, it typically exhibits an initial period of slow convergence followed by a faster decrease of the residual norms.
We concentrate on a wellknown 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 :
convectiondiffusion problem, SUPG discretization, GMRES, rate of convergence, ill conditioned eigenvectors, nonnormality,
tridiagonal Toeplitz matrices