Adaptive Triangulation Methods for Bivariate Spline Solutions of the Poisson Equation

We report numerical performance of self-adaptive triangulation approach to improve numerical solutions of Poisson equations using bivariate spline functions. Starting with a uniform triangulation, we use the gradient values of an initial spline solution to generate an updated triangulation. Both refining and coarsening the initial triangulation are combined together and a global triangulation is used instead of local modifications. Extensive numerical experiments have been conducted and will be shown. Our research results suggest that it is worthwhile to apply our self-adaptive algorithm(SAA) multiple times to gain additional accuracy. In addition, we present a heuristic for generating an initial solution dependent triangulation, and show numerical evidence that applying our algorithm to this initial guess gives better results than starting with a uniform initial mesh.