Serendipity and Tensor Product Affine Pyramid Finite Elements
The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 215-228.

Using the language of finite element exterior calculus, we define two families of H 1 -conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.

Published online:
DOI: 10.5802/smai-jcm.14
Classification: 65N30, 41A20, 41A10
Keywords: Finite element methods; pyramid elements; rational functions
Andrew Gillette 1

1 Department of Mathematics, University of Arizona, Tucson, AZ, USA 85721.
@article{SMAI-JCM_2016__2__215_0,
     author = {Andrew Gillette},
     title = {Serendipity and {Tensor} {Product} {Affine} {Pyramid} {Finite} {Elements}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {215--228},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {2},
     year = {2016},
     doi = {10.5802/smai-jcm.14},
     zbl = {1416.65445},
     mrnumber = {3633550},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.14/}
}
TY  - JOUR
AU  - Andrew Gillette
TI  - Serendipity and Tensor Product Affine Pyramid Finite Elements
JO  - The SMAI Journal of computational mathematics
PY  - 2016
SP  - 215
EP  - 228
VL  - 2
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.14/
DO  - 10.5802/smai-jcm.14
LA  - en
ID  - SMAI-JCM_2016__2__215_0
ER  - 
%0 Journal Article
%A Andrew Gillette
%T Serendipity and Tensor Product Affine Pyramid Finite Elements
%J The SMAI Journal of computational mathematics
%D 2016
%P 215-228
%V 2
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.14/
%R 10.5802/smai-jcm.14
%G en
%F SMAI-JCM_2016__2__215_0
Andrew Gillette. Serendipity and Tensor Product Affine Pyramid Finite Elements. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 215-228. doi : 10.5802/smai-jcm.14. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.14/

[1] M. Ainsworth; O. Davydov; L. Schumaker Bernstein–Bézier finite elements on tetrahedral–hexahedral–pyramidal partitions, Computer Methods in Applied Mechanics and Engineering, Volume 304 (2016), pp. 140-170 | DOI | Zbl

[2] D.N. Arnold; G. Awanou The serendipity family of finite elements, Foundations of Computational Mathematics, Volume 11 (2011) no. 3, pp. 337-344 | DOI | MR | Zbl

[3] D.N. Arnold; D. Boffi; F. Bonizzoni Finite element differential forms on curvilinear cubic meshes and their approximation properties, Numerische Mathematik (2014), pp. 1-20

[4] D.N. Arnold; R. Falk; R. Winther Finite element exterior calculus, homological techniques, and applications, Acta Numerica (2006), pp. 1-155 | DOI | MR

[5] D.N. Arnold; R.S. Falk; R. Winther Finite element exterior calculus: from Hodge theory to numerical stability, Bulletin of the American Mathematical Society, Volume 47 (2010) no. 2, pp. 281-354 | DOI | MR

[6] D.N. Arnold; A. Logg Periodic Table of the Finite Elements, SIAM News, Volume 47 (2014. femtable.org) no. 9

[7] T.C. Baudouin; J.-F. Remacle; E. Marchandise; F. Henrotte; C. Geuzaine A frontal approach to hex-dominant mesh generation, Advanced Modeling and Simulation in Engineering Sciences, Volume 1 (2014) no. 1, pp. 1-30 | DOI

[8] G. Bedrosian Shape functions and integration formulas for three-dimensional finite element analysis, International journal for numerical methods in engineering, Volume 35 (1992) no. 1, pp. 95-108 | DOI | Zbl

[9] L. Beirão Da Veiga; F. Brezzi; L.D. Marini; A. Russo Serendipity nodal VEM spaces, Computers & Fluids, Volume in press (2016) | MR | Zbl

[10] M. Bergot; G. Cohen; M. Duruflé Higher-order finite elements for hybrid meshes using new nodal pyramidal elements, Journal of Scientific Computing, Volume 42 (2010) no. 3, pp. 345-381 | DOI | MR | Zbl

[11] S. Brenner; L. Scott The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2002 | DOI | Zbl

[12] J. Chan; Z. Wang; A. Modave; T. Remacle GPU-accelerated discontinuous Galerkin methods on hybrid meshes, Journal of Computational Physics, Volume 318 (2016), pp. 142-168 | DOI | MR | Zbl

[13] J. Chan; T. Warburton A comparison of high order interpolation nodes for the pyramid, SIAM Journal on Scientific Computing, Volume 37 (2015) no. 5, p. A2151-A2170 | DOI | MR | Zbl

[14] J. Chan; T. Warburton hp-finite element trace inequalities for the pyramid, Computers & Mathematics with Applications, Volume 69 (2015) no. 6, pp. 510-517 | DOI | MR

[15] J. Chan; T. Warburton A short note on a Bernstein–Bézier basis for the pyramid, arXiv:1508.05609 (2015) | Zbl

[16] J. Chan; T. Warburton Orthogonal bases for vertex-mapped pyramids, SIAM Journal on Scientific Computing, Volume 38 (2016) no. 2, p. A1146-A1170 | DOI | MR | Zbl

[17] S. Christiansen; A. Gillette Constructions of some minimal finite element systems, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 50 (2016) no. 3, pp. 833-850 | DOI | MR | Zbl

[18] P. Ciarlet The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40, SIAM, Philadelphia, PA, 2002 | MR

[19] J-L. Coulomb; F-X. Zgainski; Y. Maréchal A pyramidal element to link hexahedral, prismatic and tetrahedral edge finite elements, IEEE Trans. Magnetics, Volume 33 (1997) no. 2, pp. 1362-1365 | DOI

[20] F. Fuentes; B. Keith; L. Demkowicz; S. Nagaraj Orientation embedded high order shape functions for the exact sequence elements of all shapes, Computers and Mathematics with Applications, Volume 70 (2015) no. 4, pp. 353 -458 | DOI | MR

[21] T. Hughes The finite element method, Prentice Hall Inc., Englewood Cliffs, NJ, 1987

[22] L. Liping; K.B. Davies; M. Krezek; L. Guan On higher order pyramidal finite elements, Advances in Applied Mathematics and Mechanics, Volume 3 (2011) no. 2, pp. 131-140 | DOI | MR | Zbl

[23] L. Liu; K. B. Davies; K. Yuan; M. Křížek On symmetric pyramidal finite elements, Dynamics of Continuous Discrete and Impulsive Systems Series B, Volume 11 (2004), pp. 213-228 | MR | Zbl

[24] J. Mandel Iterative solvers by substructuring for the p-version finite element method, Computer Methods in Applied Mechanics and Engineering, Volume 80 (1990) no. 1-3, pp. 117-128 | DOI | MR | Zbl

[25] N. Nigam; J. Phillips High-order conforming finite elements on pyramids, IMA Journal of Numerical Analysis, Volume 32 (2012) no. 2, pp. 448-483 | DOI | MR | Zbl

[26] N. Nigam; J. Phillips Numerical integration for high order pyramidal finite elements, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 46 (2012) no. 2, pp. 239-263 | DOI | Numdam | MR | Zbl

[27] G. Strang; G. Fix An analysis of the finite element method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973, xiv+306 pages

[28] B.A. Szabó; I. Babuška Finite element analysis, Wiley-Interscience, 1991

[29] F.D. Witherden; P.E. Vincent On the identification of symmetric quadrature rules for finite element methods, Computers & Mathematics with Applications, Volume 69 (2015) no. 10, pp. 1232-1241 | DOI | MR

[30] F-X. Zgainski; J-L. Coulomb; Y. Maréchal; F. Claeyssen; X. Brunotte A new family of finite elements: the pyramidal elements, IEEE Trans. Magnetics, Volume 32 (1996) no. 3, pp. 1393-1396 | DOI

Cited by Sources: