On the Convergence of Basic Iterative Methods for Convection-Diffusion Problems

J. Bey and A. Reusken
Institut f\"ur Geometrie und Praktische Mathematik
RWTH Aachen

Abstract

We consider linear systems which result from finite element or finite volume discretization of convection-diffusion problems. We analyze the convergence of basic iterative methods of Jacobi and Gauss-Seidel type applied to these linear systems. One known standard result (cf. [1]) for a class of 2D model problems uses the assumption that the underlying triangulation is of weakly acute type (the angles of the triangles are less than or equal to pi/2). The resulting matrix then is an M-matrix and a standard convergence analysis can be applied. In this talk we consider a setting in which the matrix of the discrete problem is not necessarily an M-matrix. In this setting we introduce a few weaker algebraic conditions, e.g., that the matrix is the sum of a symmetric positive definite matrix (diffusion part) and an M-matrix (convection part). Assuming that one or more of these conditions is satisfied we analyze the convergence of basic iterative methods. For a few popular finite element and finite volume methods we show which of these algebraic conditions are satisfied in general.

 

[1] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer 1996.