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Volume 34, Issue 2
The a Priori and a Posteriori Error Estimates for Modified Interior Transmission Eigenvalue Problem in Inverse Scattering

Yanjun Li, Yidu Yang & Hai Bi

DOI: 10.4208/cicp.OA-2023-0124

Commun. Comput. Phys., 34 (2023), pp. 503-529.

Published online: 2023-09

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  • Abstract

In this paper, we discuss the conforming finite element method for a modified interior transmission eigenvalues problem. We present a complete theoretical analysis for the method, including the a priori and a posteriori error estimates. The theoretical analysis is conducted under the assumption of low regularity on the solution. We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms, and we also analyze the reliability of estimators for eigenvalues. Finally, we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order. The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below, and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.

  • Keywords

Modified interior transmission eigenvalues, a priori error estimates, a posteriori error estimates, adaptive algorithm.

  • AMS Subject Headings

65N25, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-503, author = {Li , YanjunYang , Yidu and Bi , Hai}, title = {The a Priori and a Posteriori Error Estimates for Modified Interior Transmission Eigenvalue Problem in Inverse Scattering}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {2}, pages = {503--529}, abstract = {

In this paper, we discuss the conforming finite element method for a modified interior transmission eigenvalues problem. We present a complete theoretical analysis for the method, including the a priori and a posteriori error estimates. The theoretical analysis is conducted under the assumption of low regularity on the solution. We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms, and we also analyze the reliability of estimators for eigenvalues. Finally, we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order. The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below, and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0124}, url = {http://global-sci.org/intro/article_detail/cicp/21976.html} }
TY - JOUR T1 - The a Priori and a Posteriori Error Estimates for Modified Interior Transmission Eigenvalue Problem in Inverse Scattering AU - Li , Yanjun AU - Yang , Yidu AU - Bi , Hai JO - Communications in Computational Physics VL - 2 SP - 503 EP - 529 PY - 2023 DA - 2023/09 SN - 34 DO - http://doi.org/10.4208/cicp.OA-2023-0124 UR - https://global-sci.org/intro/article_detail/cicp/21976.html KW - Modified interior transmission eigenvalues, a priori error estimates, a posteriori error estimates, adaptive algorithm. AB -

In this paper, we discuss the conforming finite element method for a modified interior transmission eigenvalues problem. We present a complete theoretical analysis for the method, including the a priori and a posteriori error estimates. The theoretical analysis is conducted under the assumption of low regularity on the solution. We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms, and we also analyze the reliability of estimators for eigenvalues. Finally, we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order. The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below, and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.

Li , YanjunYang , Yidu and Bi , Hai. (2023). The a Priori and a Posteriori Error Estimates for Modified Interior Transmission Eigenvalue Problem in Inverse Scattering. Communications in Computational Physics. 34 (2). 503-529. doi:10.4208/cicp.OA-2023-0124
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