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IGPM508.pdf October 2020 
TITLE 
Analysis of finite element methods for surface vectorLaplace eigenproblems 
AUTHORS 
Arnold Reusken 
ABSTRACT 
In this paper we study finite element discretizations of a surface vectorLaplace
eigenproblem. We consider two known classes of finite element methods, namely one based on a
vector analogon of the DziukElliott surface finite element method and one based on the socalled
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 vectorLaplace 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 vectorLaplace eigenproblem. 
KEYWORDS 
vectorLaplace eigenproblem, surface finite element method, trace finite element method 