Numerical Investigations on Trace Finite Element Methods for the Laplace–Beltrami Eigenvalue Problem

In this paper, we study numerically several trace finite element methods for the Laplace–Beltrami eigenvalue problem on surfaces, including the original variant, a stabilized isoparametric element and a new method with exact geometric descriptions. The new variant is proposed directly on a smooth manifold which is implicitly given by a level-set function and require high order numerical quadrature on the surface. We show that without stabilization the eigenvalues of the discrete Laplace–Beltrami operator may coincide with only part of the eigenvalues of an embedded problem, which further corresponds to the finite eigenvalues for a singular generalized algebraic eigenvalue problem. The finite eigenvalues can be efficiently solved by a rank-completing perturbation algorithm in Hochstenbach et al. (SIAM J Matrix Anal Appl 40:1022–1046, 2019). We prove the new method has optimal convergence rate without considering the quadrature errors. The impact of the geometric consistency on the eigenvalue problem is carefully studied. Numerical experiments suggest that all the methods have optimal convergence rate while the geometric consistency can improve the numerical accuracy significantly.