A Subdivision-based Geometric Multigrid Method

Abstract

In this paper we introduce a smooth subdi- vision theory-based geometric multigrid method. While theory and efficiency of geometric multigrid methods rely on grid regularity, this requirement is often not directly fulfilled in applications where partial differential equations are defined on complex geometries. Instead of generating multigrid hierarchies with classical linear refinement, we here propose the use of smooth subdivision theory for automatic grid hierarchy regularization within a geometric multigrid solver. This subdivi- sion multigrid method is compared to the classical geometric multigrid method for two benchmark problems. Numerical tests show significant improvement factors for iteration numbers and solve times when comparing subdivision to classical multigrid. A second study fo- cusses on the regularizing effects of surface subdivision refinement, using the Poisson-Nernst-Planck equations. Subdivision multigrid is demonstrated to outperform classical multigrid.