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

TL;DR

This paper introduces a new trace finite element method for solving the Laplace-Beltrami eigenvalue problem on smooth manifolds, providing optimal convergence and improved accuracy through geometric consistency.

Contribution

The paper presents a novel trace finite element approach directly on manifolds defined by level-set functions, with comprehensive analysis and efficient eigenvalue computation techniques.

Findings

Eigenvalues of the discrete operator match part of the embedded problem's eigenvalues

The method achieves optimal convergence rates

Numerical results show significant accuracy improvements

Abstract

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.