2020/03/18 by Daniele Boffi, Boffi, Daniele, Johnny Guzmán +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2003.08381
openalex publication_date 2020/03/18 · openalex created_date 2020/03/23 · openalex updated_date 2026/07/28
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.