Fourier Analysis of Iterative Methods for Elliptic Problems

This paper presents a Fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems. As in the von Neumann stability analysis of hyperbolic and parabolic problems, the approach is easier to apply, reveals more details about convergence properties than about standard techniques, and can be applied in a systematic way to a wide class of numerical methods. Although the analysis is applicable only to periodic problems, the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions, such as the Dirichlet problem. In addition, they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the Dirichlet case.