Adaptive RBF-FD method for Poisson’s equation

Abstract

Solutions to many physical problems often vary in magnitude throughout the problem domain. Sometimes, the problematic high intensity areas are known in advance, but more often they are unknown beforehand. Adaptive techniques for solving partial differential equations 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 certain error threshold is reached. We present a recently published technique for automatic adaptivity for strong form meshless methods and to Poisson equation using the popular RBF-FD method. Both 2D and 3D cases are considered using uniform and adaptive refinement, illustrating the advantages of fully automatic adaptivity.