Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept

Abstract

In this paper, the Green's function for the annular Laplace problem is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Three cases, fixed–fixed, fixed–free and free–fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using T-complete sets. By employing the addition theorem, both solutions are found to be mathematically equivalent when the number of Trefftz base and the number of image points are both infinite. On the basis of the same number of degrees of freedom, the convergence rate of both methods is compared with each other. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner.