Volume 28, Issue 3
A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

Yingxia XiXia Ji

Commun. Comput. Phys., 28 (2020), pp. 1105-1132.

Published online: 2020-07

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  • Abstract

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.

  • Keywords

Elastic transmission eigenvalue problem, mixed finite element method, Lagrange finite element.

  • AMS Subject Headings

65N25, 65N30, 47B07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1105, author = {Xi , Yingxia and Ji , Xia}, title = {A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {3}, pages = {1105--1132}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0106}, url = {http://global-sci.org/intro/article_detail/cicp/17677.html} }
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 -

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.

Yingxia Xi & Xia Ji. (2020). A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem. Communications in Computational Physics. 28 (3). 1105-1132. doi:10.4208/cicp.OA-2019-0106
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