Numerical Study of the Almost Nested Case in a Multilevel Method Based on Non-nested Meshes

Abstract

Partial differential equations in complex domains are very flexibly discretized by finite elements with unstructured meshes. For such problems, the challenging task to construct coarse level spaces for efficient multilevel preconditioners can in many cases be solved by a semi-geometric approach, which is based on a hierarchy of non-nested meshes. In this paper, we investigate the connection between the resulting semi-geometric multigrid methods and the truly geometric variant more closely. This is done by considering a sufficiently simple computational domain and treating the geometric multigrid method as a special case in a family of almost nested settings. We study perturbations of the meshes and analyze how efficiency and robustness depend on a truncation of the interlevel transfer. This gives a precise idea of which results can be achieved in the general unstructured case. © Springer-Verlag Berlin Heidelberg 2013.