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

摘要

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