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.

在这项工作中，我们提出了一种基于Pascal多项式和多尺度方法的有效无网格方法，以解决二维（2-D）和三维（3-D）椭圆界面问题的数值解，这些问题可能具有不连续系数和弯曲带有或不带有尖角的接口。所提出的方法使用Pascal多项式作为基函数，并利用多尺度方法来稳定数值解。众所周知的事实是，由于在离散化处理之后形成的条件条件差的合成系数矩阵不佳，使用多项式基而无需进行任何修改即可获得偏微分方程的数值解。因此，作为对病态严重的系数矩阵的一种补救措施，我们采用了多尺度方法。多尺度方法背后的主要思想是使所得系数矩阵的所有列的范数彼此相等。所提出的方法是真正的强形式无网格方法，因为我们在问题领域不需要任何网格或集成过程，这些特征使得该方法在计算机环境中的实现非常简单。提出的方法的有效性通过一些测试问题进行了测试，这些问题可能具有光滑的界面或带有尖角的界面。在存在噪声效应的情况下，对所提出方法的稳定性进行了数值研究。此外，为了显示所提方法的准确性，我们与文献中可用的数值方法进行了一些比较，例如直接无网格局部Petrov-Galerkin方法，匹配界面和边界方法，谱元法和一些基于径向基函数的无网格法。所得的数值结果及其比较证实了所提出的方法在二维和3-D稳态椭圆界面问题中的适用性。