Boundary knot method based on geodesic distance for anisotropic problems

Abstract

The radial basis function (RBF) collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless, integration-free, and highly accurate. However, the RBF-based methods have markedly been limited to handle isotropic problems due to the use of the isotropic Euclidean distance. This paper makes the first attempt to use the geodesic distance with the RBF-based boundary knot method (BKM) to solve 2D and 3D anisotropic Helmholtz-type and convection-diffusion problems. This approach is mathematically simple and easy to implement, and spectral convergence is numerically observed for problems under complex-shaped boundary. Numerical results show that the BKM based on the geodesic distance can produce highly accurate solutions of anisotropic problems with a relatively small number of knots. This study provides a promising strategy for the RBF-based methods to effectively solve anisotropic problems.