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Discontinuous Galerkin Method for the Interior Transmission Eigenvalue Problem in Inverse Scattering Theory

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Abstract

In this paper, we are devoted to design the discontinuous Galerkin method to discrete the non-selfadjoint and nonlinear interior transmission eigenvalue problem. Such eigenvalues determined from scattering data provide information about material properties of the scattering media and hence can be applied to target identification and nondestructive testing. The spectral approximation of the discontinuous Galerkin method is proved and the convergence of the approximate transmission eigenvalue is at order \(O(h^{2\ell })\)(\(\ell \ge 1\)), notably observing the convergence order at \(O(h^{2\ell -2})\)(\(\ell \ge 2\)) of the finite element method and the \(C^{0}\) interior penalty Galerkin method, and at \(O(h^{2\ell -1})\)(\(\ell \ge 1\)) of the virtual element method theoretically and numerically. Representative numerical examples are implemented to demonstrate the theoretical results, including the optimal convergence on the classical triangular mesh and the developed polygonal meshes, the influence of the penalty parameter in the scheme, transmission eigenvalues on the stratified media and the inverse spectral problem.

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Data Availability

The data during the current study are available on request.

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Acknowledgements

We wish to thank the referee for his/her constructive comments and suggestions. The author thanks Dr.Yongchao Zhang from Northwestern University, **’an, China, for allowing us to use his DG codes. The work is supported by the China Scholarship Council (No.202106280167).

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  1. Department of Mathematics, College of Science, National University of Defense Technology, Changsha, 410073, Hunan, People’s Republic of China

    Jian Meng

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Meng, J. Discontinuous Galerkin Method for the Interior Transmission Eigenvalue Problem in Inverse Scattering Theory. J Sci Comput 96, 66 (2023). https://doi.org/10.1007/s10915-023-02290-7

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