TY - GEN

T1 - Hermite and Bernstein style basis functions for cubic serendipity spaces on squares and cubes

AU - Gillette, Andrew

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2014.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2014

Y1 - 2014

N2 - We introduce new Hermite style and Bernstein style geometric decompositions of the cubic serendipity finite element spaces S3(I2) and S3(I3), as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337–344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces—12 instead of 16 for the square and 32 instead of 64 for the cube—yet are still guaranteed to obtain cubic order a priori error estimates in H1 norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.

AB - We introduce new Hermite style and Bernstein style geometric decompositions of the cubic serendipity finite element spaces S3(I2) and S3(I3), as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337–344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces—12 instead of 16 for the square and 32 instead of 64 for the cube—yet are still guaranteed to obtain cubic order a priori error estimates in H1 norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.

KW - Finite elements

KW - Hermite interpolation

KW - Multivariate polynomial interpolation

KW - Serendipity elements

KW - Tensor product interpolation

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U2 - 10.1007/978-3-319-06404-8_7

DO - 10.1007/978-3-319-06404-8_7

M3 - Conference contribution

AN - SCOPUS:84927645853

T3 - Springer Proceedings in Mathematics and Statistics

SP - 103

EP - 121

BT - Approximation Theory XIV

A2 - Fasshauer, Gregory E.

A2 - Schumaker, Larry L.

PB - Springer New York LLC

T2 - 14th International conference Approximation Theory XIV, AT 2013

Y2 - 7 April 2013 through 10 April 2013

ER -