Tailored Finite Element Method For Helmholtz Equation

Solving Wave Equation in form of acoustic wave,elastic wave,electromagnetism wave, plays an important role in applied scientific research, in subjects such as electromagnetism,acoustic, seismology,material science.However,solving these wave equations in frequency-domain has turned into finding numerical solutions of Helmholtz equation. Finding numerical method of Helmholtz Equation with large wavenumber is an important topic in Scientific Computation. When solving Helmholtz equation with traditional Galerkin finite element method,there will be a bad phenomenon which called 'pollution e?ect' with the numerical solution. This e?ect is caused by the fact that mesh size is not small enough to match the wavelength,which led to a dispersion between the numerical solution and analytic solution. To eliminate 'pollution e?ect',we need to refine the mesh, then the cost of computation will increase rapidly. The main purpose of this paper is to design special a tailored FEM method for Helmholtz equation to reduce the pollution e?ect.This paper recalled finite element method of Helmholtz equaiton,the causes of'pollution e?ect' application of Helmholtz equation,and related previous works.Then we designed tailored finite element space for constant coe?cients Helmholtz equation by properties of general equation of homogeneous equations. For general variant coe?cients Helmholtz problems we designed finite element space mixed with trigonometric functions and linear functions. Our idea is came from the oscillation property of high-frequency wave.Then we proved the well-posedness of our tailored FEM method,and make related error estimation as well. We verified related theoretical results of our method by numerical examples at last.