A Finite Difference Scheme Of Helmholtz Equation Based On The Galerkin Method

Helmholtz equation is an important elliptic partial differential equation. It has the extremely widespread application in the fields of physics, electromagnetic radiation, seismology and acoustic. A large number of researches about this equation are given with different boundary. Under the previous studies, we derive a finite difference scheme based on cosine function of Helmholtz equation which combines Galerkin finite element with finite difference method in this paper.At first, we show the basic theory of the difference scheme based on variational principle as the basis for a new difference scheme. Then we construct an appropriate primary function in discrete intervals of one dimensional Helmholtz equation and then derive the difference scheme with numerical integration method and linear interpolation method. At the same time, the boundary condition and condition number are analyzed of difference scheme.The conclusion derived by one dimensional Helmholtz equation are used to construct primary functions for two-dimensional Helmholtz equation. The method of Lagrange interpolation and numerical integration are used to derive two-dimensional finite difference scheme. Discussing the Dirichlet boundary condition of difference scheme. The eigenvalue analysis and the convergence analysis are present.At last we select numerical examples to test the scheme according to boundary condition with sine transform algorithm. By comparing with the previous difference scheme, we analysis the condition number of matrix and the error of the scheme. The experimental data show that the new difference scheme is more stability and the superiority of the difference scheme is verified.