The MFS versus the Trefftz method for the Laplace equation in 3D

The method of fundamental solutions (MFS) and the Trefftz method are two powerful boundary meshless methods for solving boundary value problems governed by homogeneous partial differential equations. High accuracy can be obtained when we employ these two methods to solve equations with harmonic boundary conditions. However, dealing with equations with non-harmonic boundary conditions in irregular domains remains a challenge. Despite the long history of these two methods, each one has its disadvantages in numerical implementation. Recent advances in the Trefftz method using the multiple scale technique has made significant improvement in reducing the condition number. As a result, the Trefftz method has become more effective for solving challenging problems. Meanwhile, there has also been progress in selecting the source points in the MFS using the Leave-One-Out Cross Validation (LOOCV) method. In this paper, we propose a simple and yet effective approach to further improve the selection of source points of the MFS in 3D. Equipped with these new techniques, we compare these two methods for solving the Laplace equation with non-harmonic boundary conditions in complicated irregular domains in 3D. In this paper, we only consider the Trefftz method with cylindrical basis functions.