A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

Abstract

In this paper, we introduce a \(C^{0}\) virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming \(H^{1}(\varOmega )\times H^{1}(\varOmega )\) discrete formulation, which makes use of lower regular virtual element spaces. However, the discrete scheme is a non-classical mixed method due to the non-selfadjointness, then we cannot use the framework of classical eigenvalue problem directly. We employ the spectral theory of compact operator to prove the spectral approximation. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

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Acknowledgements

We wish to thank the referee for his/her constructive comments and suggestions. The work is supported by the Science Challenge Project (No. TZ2016002) and the Fundamental Research Funds for the Central Universities (No. xzy022019040).

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Correspondence to Liquan Mei.

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Meng, J., Wang, G. & Mei, L. A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem. Calcolo 58, 2 (2021). https://doi.org/10.1007/s10092-020-00391-5

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Keywords

  • Virtual element method
  • Polytopal mesh
  • Transmission eigenvalue problem
  • Spectral approximation

Mathematics Subject Classification

  • 65N25
  • 65N30
  • 65N15