The generalized finite difference method with third- and fourth-order approximations and treatment of ill-conditioned stars

Abstract

In this paper, we solve 2D and 3D second-order partial differential equations considering the Generalized Finite Difference Method with third- and fourth-order approximations. We analyze the influence of the number of points per star and establish some values as references. We propose a new strategy to deal with ill-conditioned stars, which are frequent in higher-order approximations. This strategy uses a few points per star and presents excellent results both for detecting ill-conditioned stars and for increasing the accuracy of the numerical approximation. We apply parallel processing to the formation of stars and derivatives, and show the speedup achieved.