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


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.