We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines.
Bibliographical noteKAUST Repository Item: Exported on 2022-04-27
Acknowledgements: D.B. is member of INdAM Research group GNCS and his research is partially supported by PRIN/MIUR and IMATI/CNR
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics
- General Mathematics