Multigrid for High-Dimensional Elliptic Equations

This paper discusses multigrid for high dimensional partial differential equations (PDEs). We present partial grid-coarsening strategies for excellent multigrid convergence in the context of elliptic PDEs. We show that the multigrid convergence rate can satisfactorily be brought down with the grid strategies proposed herein, coupled with weighted point smoothing schemes. A computer implementation of local Fourier smoothing analysis is employed to compute the optimal relaxation parameters for w-RB Jacobi in a d-dimensional setting. The use of these relaxation parameters in the smoothing process, improves convergence in higher d. We support this by the numerical experiments in the last section, demonstrating the convergence results that we get with this proposal. Multigrid ranks among one of the best known methods for the numerical solution of partial differential equations mapped to a discrete grid [1]. Problems in applied sciences, -stemming from physical systems dependent on a number of independent variables- are nowadays sometimes modelled by high dimensional partial differential equations [2,3]. This growth in the dimensionality of the problem renders many efficient algorithms impractical due to the asymptotic rise in the number of unknowns. A way around this so-called "curse of dimensionality" is the sparse grid technique. Sparse grids are always non-equidistant emphasizing the need of efficient solution methods for non-equidistant grids and this is the precise avenue where our present contribution fits in. We demonstrate (in numbers) the nice convergence that we get with these methods, for a general d-dimensional elliptic equation