The Method Of Fundamental Solution With LOOCV For Boundary Value Problems In Complicated Domains

Meshless methods are relatively new numerical methods which have gain popularity in computational and engineering sciences during the last two decades. In this paper, the method of fundamental solution (MFS) which is a simple but efficient method for boundary problems will be studied. The aim of this paper is to investigate how to efficiently select, not only the location, but also the boundary collocation points distribution, and the effect of this selection on the accuracy of the MFS.Compared with boundary element method (BEM), MFS successfully reduces the dimensionality of the original problem by one and avoids the singular numerical inte-gral via moving the source points of the fundamental solution outside the domain and approximating the solutions with a linear system of the fundamental solution. The satisfactory location of the sources outside the closure of the domain of the problem is one of the major issues in the application of MFS as the accuracy closely relate to the location of the source points. In this paper we consider four ways of choosing the boundary collocation and source points in the MFS as well as test them on various boundary value problems. Sources on the simple circle or sphere can lead to high accu-racy if the points inside the domain satisfy the boundary conditions. For the problem whose boundary conditions are not satisfied by points inside the domain, a good way to select collocation points is uniformly distributing them on the boundary. In this case, the sources are located on the normal vector that goes out of the region. We investigate the unknown parameters by means of two algorithms in this work, one based on the satisfaction of the boundary conditions and one based on the leave-one-out cross valida-tion algorithm. Optimal parameters can be obtained via minimizing the error function for problems whose exact solution is known. For general problems, it is advisable to minimize the estimate error function. By applying these algorithm to several numerical examples for the laplace and biharmonic equations in a variety of geometries in two and three dimensions, we obtain locations of the sources which lead to highly accurate results at a relatively low cost.We also investigate an image reconstruction method based on MFS. The method is a new technique for surface reconstruction from a data set of scattered points taken on a surface. In this method, the construction of a geometric model is based on the solution of an elliptic boundary value problem on a larger domain containing the original one. The three dimensional surface is constructed by selecting points satisfied with the boundary conditions.