On the convergence of basic iterative methods for convection-diffusion equations

In this paper we analyze convergence of basic iterative Jacobi and Gauss-Se idel type methods for solving linear systems which result from finite eleme nt or finite volume discretization of convection-diffusion equations on uns tructured meshes. In general the resulting stiffness matrices are neither M -matrices nor satisfy a diagonal dominance criterion. We introduce two new matrix classes and analyse the convergence of the Jacobi and Gauss-Seidel m ethods for matrices from these classes. A new convergence result for the Ja cobi method is proved and negative results for the Gauss-Seidel method are obtained. For a few well-known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright (C ) 1999 John Wiley & Sons, Ltd.