Convergence analysis of iterative methods by pseudodifference operators

The convergence of iterative methods to solve linear partial differential equations numerically is analyzed by the theory of pseudodifference operators. Approximate inverses are determined from the symbol of the iterative operator so that implicit and preconditioned methods are covered. The behavior of the symbol for higher wave numbers describes the convergence rate for the corresponding error modes. The results are applied to Krylov subspace methods, stationary iterations of Gauss–Seidel type, the multigrid algorithm and time-stepping methods for flow problems.