TY - JOUR
T1 - A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem
AU - Xi , Yingxia
AU - Ji , Xia
JO - Communications in Computational Physics
VL - 3
SP - 1105
EP - 1132
PY - 2020
DA - 2020/07
SN - 28
DO - http://doi.org/10.4208/cicp.OA-2019-0106
UR - https://global-sci.org/intro/article_detail/cicp/17677.html
KW - Elastic transmission eigenvalue problem, mixed finite element method, Lagrange
finite element.
AB - <p style="text-align: justify;">The goal of this paper is to develop numerical methods computing a few
smallest elastic interior transmission eigenvalues, which are of practical importance in
inverse 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 finite
element method which is close to the Ciarlet-Raviart mixed finite element method. The
scheme is based on Lagrange finite element and is one of the less expensive methods
in 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 mixed
problem which does not fit into the framework of classical theoretical analysis. Instead, we obtain the convergence analysis based on the spectral approximation theory
of compact operator. Numerical examples are presented to verify the theory. Both real
and complex eigenvalues can be obtained.</p>