Modified Boundary Knot Method for Multi-Dimensional Harmonic-Type Equations

Abstract

This paper presents the modified boundary knot method (MBKM) for solving the homogeneous harmonic type boundary value problems (BVPs). Since no non-singular general solutions are applicable for harmonic-type equations, the general solutions of Helmholtz-type operator with a free parameter [Formula: see text] can be used to approximate the solutions of these problems by adjusting the [Formula: see text]. Compared with the classical boundary knot method (BKM) where the source nodes are the same as the boundary collocation nodes, the MBKM employs the ghost points method, which resets the source points to a disk-like region covering the primary problem area. This modification results in better accuracy without any increase in the computational cost. On the other hand, as the accuracy of the MBKM depends heavily on the parameter [Formula: see text], the effective condition number (ECN) is first employed to find a proper [Formula: see text] in MBKM. Several 2D and 3D numerical examples are listed to illustrate the superior performance of the MBKM in solving harmonic-type BVPs. The accuracy of MBKM is improved by one to two orders of magnitude compared to the classical BKM. Meanwhile, the validity of the ECN for obtaining a suitable [Formula: see text] for problems under complex geometric regions is also demonstrated.