A triangulation algorithm for fast elliptic solvers based on domain imbedding

The following triangulation problem is considered. Let R be a rectangle, and $\Omega $ an open domain whose closure is contained in R. Consider a rectangular grid covering R. Perturb this grid by shifting points close to the boundary $\partial \Omega $ onto $\partial \Omega $. This results in a quadrilateral, almost rectangular grid. Divide each cell of this grid into two triangles along one of its diagonals. This results in a triangulation of R. A subset of this triangulation is a triangulation of an approximation $\Omega ^h $ to $\Omega $. The distance between $\partial \Omega ^h $ and $\partial \Omega $ is required to be $O(h^2 )$, where h denotes the maximum meshwidth of the rectangular grid. All triangles should be nondegenerate.Triangulations of this kind are needed for finite element domain imbedding methods for elliptic boundary value problems. A particularly simple example of such a method is reviewed. The convergence theory for this method motivates our definition of nondegeneracy of triangles.F...