Adaptive Node Refinement Collocation Method for Partial Differential Equations

Abstract

In this work, by using the local node refinement technique proposed by Behrens and Iske (2002) and Behrens et al. (2001), and a quad-tree type algorithm (Berger and Jameson, 1985; Keats and Lien, 2004), we built a global refinement technique for Kansa's unsymmetric collocation approach. The proposed scheme is based on a cell by cell data structure, which by using the former local error estimator, iteratively refines the node density in regions with insufficient accuracy. We test our algorithm for steady state partial differential equations in one and two dimensions. By using thin-plate spline kernel functions, we found that the node refinement let us to reduce the approximation error and that the node insertion is only performed in regions where the analytical solution shows a high spatial variation. In addition, we found that the node refinement outperform in accuracy and number of nodes in comparison with the global classical Cartesian h-refinement technique