A Numerical Method For Solving The Boundary Value Problem Of The Helmholtz Equation

The boundary value problems of the Helmholtz equation appear in many scientific fields and engineering applications.It is a kind of important mathematical problem often encountered in electromagnetic radiation,acoustics and seismology.There are many methods for solving the problem,such as difference method[8,9,11],finite element method[1,2],boundary element method[4,7]and etc.This paper concerns with the boundary value problem of Helmholtz equation in a two dimensional simply connected domain,and a Fourier-Bessel method is considered to solve the problems.This method was the first to be presented in[5,10]to solve the Cauchy problems for the Helmholtz equation.Based on the denseness of Fourier-Bessel function,we approximate the exact solution by linear combination of Fourier-Bessel functions,and the problems are approximated by determining the unknown coefficients in the linear combination.By using the boundary conditions,an operator equation is easily obtained,which can be solved by a regularization method since the operator is compact and injective.Thus the approximate solution of Helmholtz equation can be obtained.By deriving a lower bound for the smallest singular value of the operator,and a stable convergent regularized result can be obtained with a suitable choice of the regularization parameter.The main idea is as follows:D(?)R2 is a bounded simply connected domain with a smooth boundary.Consider the following boundary value problem△u + k2u = 0,in D,(1)(?)u/(?)v+ika=f,on aD,(2)where k>0 is the wave number,v is the unit normal to the boundary aD directed into the exterior of D,f∈ L2((?)D).Define Fourier-Bessel function φn(x)(n ∈ Z)bywhere under the polar coordinates(r,θ):x =(r cosθ,r sinθ),and the constant M>rD = maxx∈D|x|.The basic idea of the Fourier-Bessel function is to approximate the boundary value problem’s solution(1)-(2)by the linear combination of the Fourier-Bessel function of the formu(x)≈uN(x)=(?)Cnφn(x),(4)where cn(n ∈ Z)are constants.uN is an approximate solution to the Helmholtz equation.In order to determine the parameters cn(n ∈Z),by using the boundary conditions,we define the following operator equation:ANcN = f,on(?)D,(5)where AN:C2N+1→L2((?)D)defined by(ANcN)(x):= ik(?)cnφn(x)(?)/(?)v(?)cnφn(x),x∈(?)D,(6)CN＝(C0,C1,C-1,…,CN,c-N)∈C2N+1.From theorem 2.3,we know wher eN*=(c0*,C1*,C-1*…,cN*,c*-N)T∈C2N+1.ANc*N =fNε,on αD.(8)where ‖fNε-f‖L2(αD)≤C0k2τ-N+ε.The operator AN define by(5)is compact and injective from lemma 2.4,and thus the operator equation is not suitable.Consider the following perturbation equation AncδN = fδ,(9)where the perturbation data fδ∈ L2(αD)is satisfied ‖f-fδ‖L2(αD)≤δ‖f‖L2(αD),0<δ<1.A regularized solution to(9)is a linear combination of the Fourier-Bessel function:where the coefficients cnα,δ are determined by solving the following equatiolαcNαρδ+ AN*ANcNα,δ=AN*fδ(11)From theorem 2.7,we know exist positive constant C independent of N and k,such that where τ0=rex/rin.Furthermore,let η>1,τ = τ02,and τmin =rexmin/rinmax,then(1)For 0<k ≤ 1,take N = ηIn|lnδ|,and choose the regularization parameterα = k2δτ0-2N,then the following result holds‖uNα,δ-u‖H1(D)≤C(k-2δ1/2|lnδ|λ‖f‖L2(αD)+C0|lnδ|-λ+k-2|lnδ|λε),where λ =ηlnτ0.(2)For k>1,take N = 11lnk/2lnτmin+ηln|lnδ|,and choose the regularization parameterα = k-1δτ02N,then we have‖uNα,δ-u‖H1(D)≤C(kσδ1/2|lnδ|λ‖f‖L2(αD)+C0|lnδ|λ+kσ|lnδ｜λε）,(14)where σ = 7/2 + 11lnτ0/1lnτmin.In the first chapter,we introduce Helmholtz equation boundary value problem and regularization methods.The second chapter is main part of the paper,which introduces the Foirier-Bessel method for solving Helmholtz equation.We approximate the exact so-lution by the linear combination of the Fourier-Bessel function,and the operator equation is obtained by using the boundary value condition,which can be solved by regularization.Convergence and stability analysis of regularization solution are also given.The third chapter is the numerical experiment,a numerical example is given to verify the validity and stability of the Fourier-Bessel method for solving the boundary value problem of the Helmholtz equation.The fourth chapter gives the conclusion of this paper.