Analysis of finite element methods for surface vector-Laplace eigenproblems

TL;DR

This paper analyzes finite element methods for solving surface vector-Laplace eigenproblems, focusing on nonconforming discretizations, and provides error bounds and numerical validation of convergence properties.

Contribution

It introduces a general framework for nonconforming finite element discretizations of surface vector-Laplace eigenproblems and derives error bounds for eigenvalues and eigenvectors.

Findings

Error bounds depend on consistency and approximability parameters.

Numerical experiments confirm convergence properties.

Framework applies to penalized nonconforming discretizations.

Abstract

In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.