ADAPTIVE RBF-FD METHOD FOR POISSON’S EQUATION

Abstract

Solutions to many physical problems governed by partial differential equations (PDEs) often vary significantly in magnitude throughout the problem domain. Although in some special cases the areas with high error are known in advance, in general the error distribution is unknown beforehand. Adaptive techniques for solving PDEs are a standard way of dealing with this problem, where problematic regions are iteratively refined. A step further is automatic adaptivity, where problematic regions are chosen automatically using an error indicator and then refined, until a certain error threshold is reached. In this paper, we apply a recently published technique for automatic adaptivity for strong form meshless methods and solve the Poisson equation and its generalisations, using the popular RBF-FD method. Both 2D and 3D cases are considered, comparing uniform and adaptive refinement, illustrating the advantages of fully automatic adaptivity.