The MFS Versus The Trefftz Method For The Laplace Equation In 3D

The method of fundamental solutions(MFS)and the Trefftz method are two pow-erful boundary meshless methods for solving boundary value problems governed by homogeneous partial differential equations.In the Trefftz method,the approximation solution is composed by a series of T-complete functions,while in the MFS,the exact solution is approximated by a fundamental solution of differential operator.Despite the long development history and widely used in different kinds of physical areas,each method has its disadvantages in numerical implementation.The basis of Trefftz method is essentially a series of polynomial function basis,so when we use more T-complete func-tions to approximate the solution,the order of basis will exponentially increase which leads to the serious ill-conditioning of the resulting linear system equation.While the MFS needs to locate the source points on the spectra boundary outside the problem boundary to eliminate the singularity of the resulting system equation.However,the optimal location of choosing source points is a significantly challenging problem.If the source points can be properly located,then the MFS can be the most effectively meshless numerical method.Recently great advances in the Trefftz method to weaken the ill-conditioning have developed rapidly.Especially employing the multiple scale technique to reduce the condition number of resulting system equation has made obvious progress,which makes the Trefftz method more effective in solving the challenging problems.In this paper,we also employ the multiple scale technique to research the behavior of the Trefftz method in solving Laplace equation with nonharmonic boundary condition under complex three-dimensional areas.At the same time,the MFS has a great breakthrough in choosing the optimal location of source points,especially employing the Leave-One-Out Cross Validation(LOOCV)algorithm to optimize the location of source points makes the MFS a more effective numerical method.The MFS is quite effective in solving the differential equations with harmonic boundary condition.Nevertheless,it’s poor in dealing with the differential equations with nonharmonic boundary condition especially in three-dimensional problems.In this paper,we also use the LOOCV algorithm to choose the best location of source points.Meanwhile,we propose a more simple and less time consuming method to make the MFS more effective and efficient in the three-dimensional differential equation with nonharmonic boundary condition.With the new techniques,we compare these two methods on accuracy,stability,and time consuming in solving three-dimensional Laplace equation with complex areas.