A Geometrically Consistent Trace Finite Element Method For The Laplace-Beltrami Eigenvalue Problem

In this paper, we propose a new trace finite element method for the {Laplace-Beltrami} eigenvalue problem. The method 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. A comprehensive analysis for the method is provided. We show that the eigenvalues of the discrete Laplace-Beltrami operator 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 {\it Hochstenbach et al. SIAM J. Matrix Anal. Appl., 2019} \cite{hochstenbach2019solving}. We prove the method has optimal convergence rate. Numerical experiments verify the theoretical analysis and show that the geometric consistency can improve the numerical accuracy significantly.