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

Liesen, Jörg; Strakos, Zdenek

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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.