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

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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
##### Milestones
Received: 17 April 2017
Accepted: 22 July 2017
Published: 15 January 2018

Communicated by Antonia Vecchio
##### Authors
 Andrew Gillette Department of Mathematics University of Arizona Tucson, AZ United States Craig Gross Department of Mathematics University of Arizona Tucson, AZ United States Ken Plackowski Department of Mathematics University of Arizona Tucson, AZ United States