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

In this paper we analyze convergence of basic iterative Jacobi and Gauss-Seidel type of methods for solving linear systems which result from finite element or finite volume discretization of convection-diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M-matrices nor satisfy a diagonal dominance criterion. We introduce two new matrix classes and analyze the convergence of the Jacobi and GaussSeidel methods for matrices from these classes. A new convergence result for the Jacobi 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.