#### Vol. 11, No. 4, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Numerical studies of serendipity and tensor product elements for eigenvalue problems

### Andrew Gillette, Craig Gross and Ken Plackowski

Vol. 11 (2018), No. 4, 661–678
##### 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 50% 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.

##### Keywords
serendipity finite elements, eigenvalue approximation, $h$-refinement, $p$-refinement
##### Mathematical Subject Classification 2010
Primary: 35P15, 41A25, 65H17, 65N30