A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem (Communications in Computational Physics, 2020)
The goal of this paper is to develop numerical methods computing a fewsmallest elastic interior transmission eigenvalues, which are of practical importance ininverse elastic scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. In this paper, we construct a lowest-order mixed finiteelement method which is close to the Ciarlet-Raviart mixed finite element method. Thescheme is based on Lagrange finite element and is one of the less expensive methodsin terms of the amount of degrees of freedom. Due to the non-self-adjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixedproblem which does not fit into the framework of classical theoretical analysis. Instead, we obtain the convergence analysis based on the spectral approximation theoryof compact operator. Numerical examples are presented to verify the theory. Both realand complex eigenvalues can be obtained.
Keywords: Elastic transmission eigenvalue problem; mixed finite element method; Lagrange finite element.