The Serendipity Family of Finite Elements

Douglas N. Arnold; Gerard Awanou

doi:10.1007/s10208-011-9087-3

The Serendipity Family of Finite Elements

Douglas N. Arnold, Gerard Awanou

Mathematics

Research output: Contribution to journal › Article › peer-review

78 Scopus citations

Abstract

We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.

Original language	English (US)
Pages (from-to)	337-344
Number of pages	8
Journal	Foundations of Computational Mathematics
Volume	11
Issue number	3
DOIs	https://doi.org/10.1007/s10208-011-9087-3
State	Published - Jun 2011

Bibliographical note

Funding Information:
The work of G. Awanou was supported in part by NSF grant DMS-0811052 and the Sloan Foundation.

Funding Information:
The work of D.N. Arnold was supported in part by NSF grant DMS-0713568.

Keywords

Finite element
Serendipity
Unisolvence

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10.1007/s10208-011-9087-3

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N2 - We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.

AB - We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.

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The Serendipity Family of Finite Elements

Abstract

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