Multigrid for high-order accurate difference equations on overset grids

Abstract

Multigrid algorithms of solving elliptic or hyperbolic partial differential equations and boundary value problems to high-order accuracy on overset grids are developed and studied in this thesis. The thesis consists of three main parts. In the first part, we consider several nonstandard coarsening strategies for geometric multigrid, including lower-order accurate coarse-level operators, red-black coarsening, and general factor-r coarsening. In the second part, the fourth-order accurate multigrid algorithm on overset grids is described, for solving elliptic problems with general geometry in two and three dimensions. The algorithm is implemented in the Ogmg solver in the Overture framework. Theoretical results based on local Fourier analysis and model problems are effectively applied on overset grids, to optimize multigrid convergence. In the third part, we examine the the application of multigrid solvers to high-order accurate implicit schemes for the wave equation.