A Priori Error Estimates for Finite Element Approximations to Eigenvalues and Eigenfunctions of the LaplaceBeltrami Operator
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 LaplaceBeltrami operator. We present and analyze approximations by Surface Finite Element Methods (SFEM) of the LaplaceBeltrami 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1801.00197
 Bibcode:
 2018arXiv180100197B
 Keywords:

 Mathematics  Numerical Analysis