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PreprintarXiv licencearXiv

Isogeometric analysis for the Helmholtz transmission eigenvalue problem

  • Nizheng Liao
  • Guanghui Hu
  • Xucheng Meng
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Abstract

The transmission eigenvalue problem plays an increasingly important role in inverse scattering theory. Although significant progress has been made in developing efficient numerical methods for the problem over the last two decades, its numerical treatment for curved domains in ℝd (d=2,3) remains challenging. In this paper, we introduce and analyze a geometrically flexible and H2-conforming isogeometric method for solving a fourth-order, quadratic and non-self-adjoint eigenvalue problem arising from the Helmholtz transmission eigenvalue problem. Using the spectral approximation theory for compact non-self-adjoint operators, we derive optimal error estimates for the discrete eigenvalues and eigenfunctions. Numerical results for the two- and three-dimensional benchmark problems, including those defined in curved domains, are presented to verify our theoretical results, and to demonstrate the advantages of the method over existing numerical methods in terms of both accuracy and geometric flexibility.

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Cite

BibTeXRIS
BibTeX
@misc{liao2026isogeometric,
  title = {{Isogeometric analysis for the Helmholtz transmission eigenvalue problem}},
  author = {Nizheng Liao and Guanghui Hu and Xucheng Meng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21862},
  eprint = {2609.21862},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21862},
}
RIS
TY  - UNPB
TI  - Isogeometric analysis for the Helmholtz transmission eigenvalue problem
AU  - Nizheng Liao
AU  - Guanghui Hu
AU  - Xucheng Meng
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21862
UR  - https://arxiv.org/abs/2609.21862
AB  - The transmission eigenvalue problem plays an increasingly important role in inverse scattering theory. Although significant progress has been made in developing efficient numerical methods for the problem over the last two decades, its numerical treatment for curved domains in ℝ^d (d=2,3) remains challenging. In this paper, we introduce and analyze a geometrically flexible and H^2-conforming isogeometric method for solving a fourth-order, quadratic and non-self-adjoint eigenvalue problem arising from the Helmholtz transmission eigenvalue problem. Using the spectral approximation theory for compact non-self-adjoint operators, we derive optimal error estimates for the discrete eigenvalues and eigenfunctions. Numerical results for the two- and three-dimensional benchmark problems, including those defined in curved domains, are presented to verify our theoretical results, and to demonstrate the advantages of the method over existing numerical methods in terms of both accuracy and geometric flexibility.
ER  -