On Multigrid for High-Dimensional Anisotropic Partial Differential Equations

H. bin Zubair

h.binzubair@tudelft.nl

Numerieke Wiskunde, DIAM, TUDelft

The Netherlands

Robust and efficient solution techniques are developed for high-dimensional elliptic partial differential equations (PDEs). High dimensional equations are often treated by the sparse-grid technique which gives rise to a number of smaller sized problems discretized on non-equidistant grids. Such discretization induces a discrete anisotropy in the system. We develop a general and robust multigrid method for high dimensional anisotropic equations with general anisotropies as well as discrete grid-induced anisotropies. In the talk we describe the d-dimensional multigrid components and operators based on Kronecker-tensor products of low dimensional operators. We show how optimal relaxation parameters can be computed for the model d-dimensional problem. Multigrid efficiency in d-dimensions depends on optimal relaxation and ideal coarse-grid correction to quite some extent, and (as opposed to the model problem realm) there exist applications where these optimal attributes may not be accessible. In such a situation, we employ our d-multigrid method as a Bi-CGSTAB preconditioner instead of a stand-alone solver, and demonstrate that this is a close substitution of optimality in the multigrid process. The resulting solver converges well for a wide class of discrete high dimensional problems.

In the talk we skim briefly through the model problem, the Sparse-Grid combination method, Bi-CGSTAB and d-multigrid. This is followed by numerical experiments conducted on a variety of equidistant and non-equidistant grids. Convergence diagrams of these experiments are displayed with author opinion about the choice of multigrid attributes and parameters. Lastly, we draw some conclusions from the work and indicate related areas of interest for future work.