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.