Local meshless methods for second order elliptic interface problems with sharp corners

Abstract

In the present paper, we develop a local meshless procedure for solving a steady state two-dimensional interface problem having discontinuous coefficients and curved interfaces with sharp corners. The proposed local meshless methods are based on three types of radial basis functions (RBFs): a local meshless method based on multiquadric RBF (LMM1P), a local meshless method based on integrated multiquadric RBF (LMM2P) and a local meshless method based on hybrid Gaussian-Cubic RBF (LMM3P). Stencils are designed at the interface and interior regions to cope with discontinuities and sharp corners. Due to the localized nature of the procedure and a sparse matrix representation, the local meshless methods become computationally less expensive than global meshless methods. The methods are augmented with linear polynomial to improve accuracy and ensure stable computation. Comparison with some existing versions of finite element methods is also performed to show better accuracy of the proposed meshless methods. Accuracies of the proposed local meshless methods are also compared among themselves. Flexibility of the meshless methods with respect to complex geometries, adapting to different shapes of the interfaces and selection of the shape parameter is also considered.