Multilevel methods for elliptic problems on domains not resolved by the coarse grid

Abstract

Elliptic boundary value problems are frequently posed on complicated domains, which cannot be covered by a simple coarse initial grid as is needed for multigrid-like iterative methods. In the present article, this problem is resolved for selfadjoint second order problems and Dirichlet boundary conditions. The idea is to construct appropriate subspace decompositions of the corresponding finite element spaces by way of an embedding of the domain under consideration into a simpler domain like a square or a cube. Then the general theory of subspace correction methods can be applied.