Numerical Algorithms 21 (1999) 387–392 387 Coarse grid spaces for domains with a complicated boundary Harry Yserentant Mathematisches Institut der Universit¨ at T¨ ubingen, D-72076 T¨ ubingen, Germany E-mail: yserentant@na.uni-tuebingen.de It is shown that, with homogeneous Dirichlet boundary conditions, 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. Keywords: multigrid methods, domains of complicated shape AMS subject classification: 65N55, 65N30 1. Introduction Multigrid methods notoriously depend on the existence of a hierarchy of grids. This can cause problems for domains that cannot be well represented by a simple initial grid with few nodes because it is generally believed that the coarse grid equations need to be solved exactly. If the coarse grid contains too many elements, the advantage of multigrid methods lessens. Many different approaches to overcome this difficulty have been proposed during the last years. Most of them are based on coarsening strategies for unstructured meshes. A very elaborate method of this kind is due to Bank and Xu [2]. More recently, Bank and Smith developed the multigraph iterative methods [1], which are a generalization of the hierarchical basis multigrid method and which Bank implemented very successfully in his finite element package PLTMG. Braess [3] and Mandel and Vanek [6] propose agglomeration techniques. Kornhuber and Yserentant [5] and later Xu [8] use subspaces from a hierarchy of finite element spaces associated with a simple domain covering the given domain to build up multigrid-like iterative methods. The approach of Hackbusch and Sauter [4] starts from a similar idea and is particularly directed to Neumann-type boundary conditions. The present paper has been inspired by both the work of Kornhuber and Yserentant, and of Hackbusch and Sauter. The at first glance surprising observation is made that, with homogeneous Dirichlet boundary conditions, the spectral condition of a finite element discretization matrix remains uniformly bounded independent of the size of the boundary elements as long as only the size of the elements increases with  J.C. Baltzer AG, Science Publishers