Numerical studies of serendipity and tensor product elements for eigenvalue problems

TL;DR

This paper demonstrates through numerical experiments that serendipity finite elements can achieve comparable accuracy to tensor product elements in eigenvalue problems while significantly reducing degrees of freedom, promoting their use.

Contribution

It provides the first detailed numerical comparison of serendipity and tensor product elements for eigenvalue problems, highlighting efficiency gains and offering basis function tables and generation tools.

Findings

Serendipity elements attain similar accuracy with fewer basis functions.

Serendipity methods can use half as many basis functions as tensor product methods.

Tools for generating basis functions are provided for further research.

Abstract

While the use of finite element methods for the numerical approximation of eigenvalues is a well-studied problem, the use of serendipity elements for this purpose has received little attention in the literature. We show by numerical experiments that serendipity elements, which are defined on a square reference geometry, can attain the same order of accuracy as their tensor product counterparts while using dramatically fewer degrees of freedom. In some cases, the serendipity method uses only half as many basis functions as the tensor product method while still producing the same numerical approximation of an eigenvalue. To encourage the further use and study of serendipity elements, we provide a table of serendipity basis functions for low order cases and a Mathematica file that can be used to generate the basis functions for higher order cases.