A generalized predictive analysis tool for multigrid methods

Multigrid and related multilevel methods are the approaches of choice for solving the linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the wellknown local mode (often local Fourier) analysis approach to a wider class of problems. The semi-algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two-dimensional convection-diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short-term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright c © 0000 John Wiley & Sons, Ltd.