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A \(C^{1}-C^{0}\) conforming virtual element discretization for the transmission eigenvalue problem

Abstract

In this study, we present and analyze a virtual element discretization for a nonselfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Using suitable projection operators, the sesquilinear forms are discretized by only using the proposed degrees of freedom associated with the virtual spaces. Under standard assumptions on the polygonal meshes, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal-order error estimate for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

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Acknowledgements

The first author was partially supported by the National Agency for Research and Development, ANID-Chile, through FONDECYT project 1180913 and by project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal.

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Correspondence to David Mora.

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Mora, D., Velásquez, I. A \(C^{1}-C^{0}\) conforming virtual element discretization for the transmission eigenvalue problem. Res Math Sci 8, 56 (2021). https://doi.org/10.1007/s40687-021-00291-2

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Keywords

  • Transmission eigenvalues
  • Spectral problem
  • Virtual element method
  • Polygonal meshes
  • Error estimates

Mathematics Subject Classification

  • 35P30
  • 65N15
  • 65N25
  • 65N30
  • 78A46