Abstract
We consider the family of finite element spaces of fixed dimension, defined over all triangulations of a given polygonal domain in R 2, based on a common set of nodes (vertices). A minimum energy triangulation, relative to a given elliptic problem, is a triangulation for which the finite element solution has the minimal energy. The minimum energy triangulation can be considered optimal, as it minimizes the solution error in the natural norm associated with the problem. It is shown in this paper that the well-known Delaunay triangulation of a set of points in R 2 is a minimum energy triangulation for the energy functional associated with the nonhomogeneous Laplace equation. Minimum energy triangulations may be very expensive to compute. In this paper, therefore, we present algorithms for constructing locally minimal energy triangulations and outline efficient schemes for computing sub-optimal triangulations. In both cases the basic idea is to improve an initial triangulation by using local operations on the edges of the triangulation. It is shown for several model problems that such sub-minimal energy triangulations can significantly improve the quality of the approximate solution.
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