A High-Order Radial Basis Function Finite Difference Method for Linear Anisotropic Elliptic Interface Problems

Tao Cheng

doi:10.1007/s10915-026-03330-8

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A High-Order Radial Basis Function Finite Difference Method for Linear Anisotropic Elliptic Interface Problems

2026-05-19

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Tao Cheng

Lili Ju *

Xiaoping Zhang

DOI：10.1007/s10915-026-03330-8

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Abstract

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Anisotropic elliptic interface problems are prevalent in various scientific and engineering disciplines. Traditional mesh-based numerical methods often struggle with complex geometries near interfaces, which can significantly impact their performance. In contrast, meshless methods, which eliminate the need for mesh generation, offer inherent advantages in handling interface problems and have emerged as a promising alternative. In this paper, we propose a high-order radial basis function finite difference (RBF-FD) method, which is specially designed to solve second-order linear anisotropic elliptic interface problems. The main idea of our method is to utilize the domain decomposition associated with the interfaces and construct the RBF-FD approximations of functions and differential operators in each of the subdomains independently to resolve inherent discontinuities on the interfaces. Detailed discretization schemes for the proposed method are presented together with implementation algorithm. Extensive numerical experiments on various interfaces are also performed to demonstrate high-order accuracy and computational efficiency of the proposed RBF-FD method.

Keywords:

Linear elliptic interface problem

Anisotropic tensor

Radial basis function

Finite difference

Subdomain-based stencil