A Mixed Method For Helmholtz Transmission Eigenvalue Problem

Helmholtz transmission eigenvlaue problem is a hotspot in mathematics and mechanics community because it has widely applications in materials science and many scholars pay attentions on the numerical treatment of this problem.It is difficult to solve because it is a non-elliptic and non-selfadjoint problem and its knowledge system is not covered by the standard theory of eigenvalue problems.At this field,Cakoni,Monk and Sun bring forth a new method for the weak formulation and make the error analysis for transmission eigenvalues of the finite element approximation.Meanwhile,Yang,Bi,Li and Han use Ciarlet-Raviart mixed method for this problem and get the wonderful effect.Because of the wide application of the mixed element method for the biharmonic equation,the study of mixed element method which can reduce the space freedom and improve the calculation efficiency is of great value.Based on the research methods and achievements of many scholars,a mixed method is proposed by us to solve the eigenvalue problem.We prove the correctness and feasibility of the method using special demonstration,and the expected effect is achieved.First of all,we combine the formula proposed by Cakoni et al with Ciarlet-Raviart mixed method,and propose a new mixed variational formulation.We prove the existence and uniqueness of the solution of the mixed variational formula by two regularity estimates and the corresponding conjugate problem is given.Then we prove that the solution operators of the variational problem and the discrete problem are compact operators.We define the Lagrange interpolation operator and Newman projection operator specially.Using interpolation theory,we prove the convergence of the finite element solution of the problem,and then we deduce that the solution operator of discrete problem converges to the one of variational problem.Based on this conclusion,we can rely on the classical eigenvalue theory to illustrate the convergence of eigenvalues and eigenfunctions of the mixed method.At last,the convergence order of the problem can be further improved by special method which illustrates the feasibility and efficiency of mixed method.