Analysis For Method Of Fundamental Solutions For The Helmholtz Equation

Helmholtz equation is an important mathematical physical equation,it usually ap-pears in study of physical problems involved partial differential equations that depend on both space and time simultaneously.Transmission of wave,control of noise,vibration of membrane and many other problems are governed by the Helmholtz equation.The method of fundamental solutions(MFS)is a method for the solution of certain elliptic boundary value problems,which may be viewed as an indirect boundary element method.In the MFS,the solution is approximated by a set of fundamental solutions of the governing equations which are expressed in terms of sources located outside the domain of the problem.The unknown coefficients in the linear combination of the fundamental solutions and the final locations of the sources are determined so that the boundary conditions are satisfied in a least squares sense.In this thesis,we study the Helmholtz equation by the MFS using Bessel and Neu-mann functions.We mainly focus on the analysis of error and stability.The bounds of errors are derived for bounded simply-connected domains,while the bounds of condition number are derived only for disk domains.There are three novelties in this thesis.The first is that we find that the MFS using Bessel functions is more efficient than the MFS using Neumann functions.After analyzing carefully,we note that by using Bessel functions,the radius R of the source nodes is not necessarily to be larger than the maximal radius rax of the solution domain.This is against the well-known rule:rmax<R for the MFS.Numerical experiments are carried out,to support the theoretic analysis and conclusions made.This is the first novelty in this thesis.The error analysis for the Helmholtz equation is more complicated than that for the modified Helmholtz equation in[1],since the Bessel functions Jn(x)have infinite zeros.We consider the curial and degenerate cases when Jn(kR)? 0 and Jn(kp)? 0.There exist few reports for the analysis for such a degeneracy(e.g.,Li[2])case.The error bounds are also explored for bounded simply-connected domains.So,the second novelty of this thesis is for the analysis of the MFS in degeneracy.For the MFS using Neumann functions,the rule of the MFS,rmax<R,must obey.For the method of particular solutions(MPS)in[3],however,the source nodes disappear.In this thesis,we will briefly provide the analysis of the MFS using Neumann functions and show that the polynomial convergence can be achieved for bounded simple-connected domains.The error analysis of the MFS using Neumann functions and numerical com-parisons for different methods are the third contribution in this thesis.In addition,this thesis discovers that the MFS using Bessel and Neumann functions suffer from the spurious eigenvalues.The spurious eigenvalues are not the true eigenvalues of the corresponding eigenvalue problems,but the correct solutions can not be obtained due to either algorithm singularity or divergence of numerical solutions.