Vol. 11, No. 4, 2018

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
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