Multigrid for High-Dimensional Elliptic Partial Differential Equations on Non-equidistant Grids

This work presents techniques, theory and numbers for multigrid in a general ddimensional setting. The main focus is the multigrid convergence for high-dimensional partial differential equations (PDEs). As a model problem we have chosen the anisotropic diffusion equation, on a unit hypercube. We present some techniques for building the general d-dimensional adaptations of the multigrid components and propose grid coarsening strategies to handle anisotropies that occur due to discretization on a non-equidistant grid. Apart from the practical formulae and techniques, we present the smoothing analysis of point ω-Red-Black Jacobi method for a general multidimensional case. We show how relaxation parameters may be evaluated and used for better convergence. This analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as a second and a fourth order finite difference operator. Finally we present some results derived from numerical experiments based on the test problem.