Symmetric boundary knot method

TL;DR

This paper introduces a symmetric boundary knot method (BKM) for solving PDEs, enhancing the original BKM by maintaining symmetry even with mixed boundary conditions, and validates its accuracy and efficiency through numerical experiments.

Contribution

A symmetric version of the boundary knot method is developed, improving its applicability and performance for complex PDE problems with mixed boundary conditions.

Findings

The symmetric BKM maintains accuracy in 2D and 3D problems.

It is meshfree, superconvergent, and easy to implement.

Numerical tests confirm its efficiency and robustness.

Abstract

The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general PDEs. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a nonsingular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, superconvergent, integration free, very easy to learn and program. The original BKM, however, loses symmetricity in the presense of mixed boundary. In this study, by analogy with Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion reaction problems under complicated geometries.