An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems

Abstract

In this work, we propose an efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solutions of two-dimensional (2-D) and three-dimensional (3-D) elliptic interface problems which may have discontinuous coefficients and curved interfaces with or without sharp corners. The proposed method uses Pascal polynomials as basis functions and utilizes multiple-scale approach for stabilizing numerical solutions. It is a well known fact that using polynomial basis without any modifications to obtain numerical solutions of partial differential equations may be peculiar owing to ill conditioned resultant coefficient matrix which is formed after process of discretization. Hence, as a remedy to the highly ill-conditioned coefficient matrix we employ multiple-scale approach. The main idea behind the multiple-scale approach is to make norm of all columns of resultant coefficient matrix equal to each other. The proposed method is a truly strong-form meshfree method since we do not need any mesh or integration process in problem domain, these features makes the implementation of the method very simple in computer environment. The efficiency of the proposed method is tested by some test problems which may have smooth interface or interface with sharp corners. Stability of the proposed method is investigated numerically in the presence of noise effect. Further, to show accuracy of the proposed method we present some comparisons with available numerical methods in literature, such as direct meshless local Petrov-Galerkin method, matched interface and boundary method, spectral element method and some meshless methods based on radial basis functions. The obtained numerical results and their comparisons confirm applicability of the proposed method for 2-D and 3-D steady state elliptic interface problems.