Numerical Solution For Elmholtz Transmission Eigenvalue Problems

In this paper,we develop a shift-and-invert algorithm for the computation of some eigenvalues nearby a given number ? and the corresponding eigenfunctions of the Helmholtz transmission eigenvalue problem.Firstly,by using the continuous finite element method,the Helmholtz transmission eigenvalue problem is discreted into a generalized eigenval-ue problem(GEP).Then we derive an associated quadratic eigenvalue problem(QEP)which eliminates the nonphysical zero eigenvalues while preserving all nonzero ones.By linearization we transform the QEP into a new GEP,and we use the shift-and-invert technique and QZ method to solve the new GEP.The proposed algorithm has no special restrictions to the refractive index of the transmission eigenvalue problems,that is to say that the refractive index can be positive or negative constant or a real function.The final numerical examples verify the effectiveness of our algorithm.This paper is organized as follows.The first chapter introduces the origin of the transmission eigenvalue problems,and some physical background,illustrates the prede-cessors' work as well as the main direction of this paper.The second chapter discretes the transmission eigenvalue problem with finite element method(FEM)into a generalized eigenvalue problem,and derives an associated quadratic eigenvalue problem.The third chapter presents the specific numerical method.The fourth chapter gives some results of numerical experiments.