A Priori Error Estimates for Finite Element Approximations to Eigenvalues and Eigenfunctions of the Laplace-Beltrami Operator

TL;DR

This paper analyzes the errors in finite element approximations of Laplace-Beltrami eigenvalues and eigenfunctions on surfaces, highlighting how geometric and Galerkin errors interact and how interpolation choices affect convergence.

Contribution

It provides a detailed error analysis for Surface Finite Element Methods applied to the Laplace-Beltrami eigenvalue problem, revealing new insights into geometric error behavior.

Findings

Eigenfunction approximation errors are influenced by surface and finite element errors.

Eigenvalue errors depend on the interpolation points used for surface approximation.

Geometric consistency error can be reduced faster by choosing appropriate interpolation points.

Abstract

Elliptic partial differential equations on surfaces play an essential role in geometry, relativity theory, phase transitions, materials science, image processing, and other applications. They are typically governed by the Laplace-Beltrami operator. We present and analyze approximations by Surface Finite Element Methods (SFEM) of the Laplace-Beltrami eigenvalue problem. As for SFEM for source problems, spectral approximation is challenged by two sources of errors: the geometric consistency error due to the approximation of the surface and the Galerkin error corresponding to finite element resolution of eigenfunctions. We show that these two error sources interact for eigenfunction approximations as for the source problem. The situation is different for eigenvalues, where a novel situation occurs for the geometric consistency error: The degree of the geometric error depends on the choice of interpolation points used to construct the approximate surface. Thus the geometric consistency term can sometimes be made to converge faster than in the eigenfunction case through a judicious choice of interpolation points.