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This work presents techniques, theory, and numbers for multigrid in a general $d$-dimensional setting. The main focus of this paper is the multigrid convergence for high-dimensional partial differential equations on non-equidistant grids such as may be encountered in a sparse-grid solution. As a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular 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 are induced due to discretization on a non-equidistant grid. Apart from the practical formulas and techniques, we present—in detail—the smoothing analysis of the point $\omega$-red-black Jacobi method for a general multidimensional case. We show how relaxation parameters may be evaluated efficiently and used for better convergence. This analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second- and the fourth-order finite difference operators. Finally we present some results derived from numerical experiments based on the test problem.

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