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Volume 32, Issue 2
A Holomorphic Operator Function Approach for the Transmission Eigenvalue Problem of Elastic Waves

Yingxia Xi & Xia Ji

DOI: 10.4208/cicp.OA-2022-0050

Commun. Comput. Phys., 32 (2022), pp. 524-546.

Published online: 2022-08

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  • Abstract

The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method. To use the abstract approximation theory for holomorphic operator functions, we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator. The spectral indicator method is employed to compute the transmission eigenvalues. Extensive numerical examples are presented to validate the theory.

  • Keywords

Discontinuous Galerkin method, transmission eigenvalue problem, elastic waves, Fredholm operator.

  • AMS Subject Headings

65N25, 65N30, 47B07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-524, author = {Xi , Yingxia and Ji , Xia}, title = {A Holomorphic Operator Function Approach for the Transmission Eigenvalue Problem of Elastic Waves}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {2}, pages = {524--546}, abstract = {

The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method. To use the abstract approximation theory for holomorphic operator functions, we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator. The spectral indicator method is employed to compute the transmission eigenvalues. Extensive numerical examples are presented to validate the theory.

}, issn = {1991-7120}, doi = {https://doi.org/ 10.4208/cicp.OA-2022-0050}, url = {http://global-sci.org/intro/article_detail/cicp/20867.html} }
TY - JOUR T1 - A Holomorphic Operator Function Approach for the Transmission Eigenvalue Problem of Elastic Waves AU - Xi , Yingxia AU - Ji , Xia JO - Communications in Computational Physics VL - 2 SP - 524 EP - 546 PY - 2022 DA - 2022/08 SN - 32 DO - http://doi.org/ 10.4208/cicp.OA-2022-0050 UR - https://global-sci.org/intro/article_detail/cicp/20867.html KW - Discontinuous Galerkin method, transmission eigenvalue problem, elastic waves, Fredholm operator. AB -

The paper presents a holomorphic operator function approach for the transmission eigenvalue problem of elastic waves using the discontinuous Galerkin method. To use the abstract approximation theory for holomorphic operator functions, we rewrite the elastic transmission eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for the discontinuous Galerkin method is proved following the abstract theory of the holomorphic Fredholm operator. The spectral indicator method is employed to compute the transmission eigenvalues. Extensive numerical examples are presented to validate the theory.

Xi , Yingxia and Ji , Xia. (2022). A Holomorphic Operator Function Approach for the Transmission Eigenvalue Problem of Elastic Waves. Communications in Computational Physics. 32 (2). 524-546. doi: 10.4208/cicp.OA-2022-0050
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