Local Meshless Method For Elliptic Interface PDEs Via RBFs Augmented with Pascal Polynomials

Abstract

The solution of elliptic partial differential equations (PDEs) with sharp interfaces presents significant challenges, especially when the domain is irregular or complex, as gradients may not be well-defined at corners and may exhibit discontinuous derivatives. The local meshless method is a promising approach for addressing such difficulties. In [1], a Radial Basis Function collocation technique was proposed for the solution of 2nd-order elliptic interface PDEs having sharp corners, using stencils at the interface designed to handle corner discontinuities. While this method demonstrated notable improvements over conventional approaches, there are instances where accuracy falls short of expectations. In this work, we enhance the local RBF method by incorporating Pascal polynomials, aiming to improve numerical solutions for elliptic interface PDEs. We evaluate the method’s effectiveness in handling complex geometries and its adaptability to various interface shapes