Coarse Grid Spaces for Domains with a Complicated Boundary

Harry Yserentant
Mathematisches Institut der Universität Tübingen,
Auf der Morgenstelle 10,
D--72076 Tübingen, Germany\\


Elliptic boundary value problems are often posed on complicated domains which cannot be covered by a simple coarse initial grid as it is needed for classical multigrid-like iterative methods where the coarse grid equations are solved exactly. Several solutions to this problem are presented for the case of homogeneous Dirichlet boundary conditions. The first technique is to construct appropriate subspace decompositions by way of an embedding of the domain under consideration into a square or a cube. The second technique is even simpler. It is shown that the condition number of finite element discretization matrices remains uniformly bounded independent of the size of the boundary elements provided that the size of the elements increases with their distance to the boundary. This fact allows the construction of simple multigrid methods of optimal complexity for domains of nearly arbitrary shape.