A global RBF-QR collocation technique for solving two-dimensional elliptic problems involving arbitrary interface

Abstract

In recent years, several numerical methods have been developed for differential equations with discontinuous coefficients, which are known as interface problems. Our aim is to introduce an efficient global meshless method for solving elliptic problems with arbitrary interface. Meshless methods based on infinitely smooth radial basis functions (RBFs) can achieve spectral convergence rate for solving partial differential equations. RBF-QR is a newly developed method which allows numerically stable computations with RBFs for all values of the free shape parameter $$\varepsilon$$. In this paper, an appropriate RBF-QR collocation technique is applied for two-dimensional elliptic interface problems, including Poisson, Helmholtz and elasticity problems. The Helmholtz interface problem is studied, by taking into account the fact that the solutions of such problems might be oscillatory. The elasticity interface problem is discussed as an application of elliptic interface problem in vector form. Several problems involving complex interface geometries and discontinuous variable coefficients are presented to confirm the efficiency and stability of RBF-QR method for this class of problems in the case of arbitrary small values of $$\varepsilon$$.