Spectral Element Method For Helmholtz Transmission Eigenvalue Problems

The transmission eigenvalue problem of the Helmholtz equation is an important part of the inverse scattering theory,and it not only has important theoretical value,but also plays a key role in some practical problem.This is due to the fact that transmission eigenvalues can be determined from the far field data of the scattered wave and used to obtain for the material properties of the scattering object.More and more scholars have paid attention to the study of this.In this paper,we study the transmission eigenvalue problem in the case of scattering of acoustic waves by a bounded simply connected inhomogenous medium.In order to solve the transmission eigenvalue more accurately and more efficiently,the spectral element method based on Chebyshev interpolation base function is presented.This method combines the flexibility of boundary and region processing with finite element and the fast convergence of spectral methods.The main work and innovations of the transmission eigenvalue are summarized as follows:Firstly,the spectral element method is studies for solving the problem of regular regions.Based on the nonlinear and non-self-adjoint properties of the transmission eigenvalue problem,the original problem is transformed into a class of quadratic eigenvalue problems by weighted residual method.Considering that the traditional finite element method is insufficient in accuracy,the Chebyshev spectral element method selects Chebyshev polynomial extremum points in the entire cell area to construct a global base function,which improves the convergence speed of the solution represented by the series,and it has a better advantage than the triangular unit when calculating the integral.Parallel compution method was introduced in the specific programming solution.The numerical experiments show the effectiveness of the Chebyshev spectral element method.Compared with the traditional finite element method,it explained the advantages of the calculation accuracy and the running speed.Secondly,an isoparametric spectral element method is proposed to solve the eigenvalue problem of Helmholtz equations in an irregular region.Based on the inadaptability of rectangular elements to irregular regions,especially curved regions,the solution of equalparameter units was introduced.By using an encryption grid,irregular regions can be well approximated and the exact solution can be maximized.Given the coordinate transformation matrix of the isoparametric unit and the Jacobian matrix,the mathematical expressions of the stiffness matrix and the mass matrix of the quadratic eigenvalue problem are derived,which has a better integral form than the traditional finite element method.Selecting a variety of irregular regions to perform numerical experiments shows that the isoparametric spectral element method has a strong regional adaptability for solving the problem of transmission eigenvalues.Compared with the finite element method,the computational efficiency is greatly improved.Finally,we summarize the full text,give the innovative points of this paper's research work,and give a prospect to the future research direction.