Computational Serendipity and Tensor Product Finite Element Differential Forms

Gillette, Andrew; Kloefkorn, Tyler; Sanders, Victoria

The SMAI Journal of computational mathematics, Tome 5 (2019), pp. 1-21.

Many conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a “computational basis” of shape functions for element families. The tensor product family, ${𝒬}_r^{-}{\Lambda }^k$, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, ${𝒮}_r^{-}{\Lambda }^k$ and ${𝒮}_r{\Lambda }^k$ respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order $r\ge 1$ and for any differential form order $0\le k\le n$, for problems in dimension $n=2$ or $3$. The bases are defined via shared subspace structures, allowing easy comparison of elements across families. We use and include code in SageMath to find, list, and verify these computational basis functions.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

• supp_02.pdf

• construct-tools-n3.sage

Classification : 65N30
Mots clés : Finite element differential forms, finite element exterior calculus, serendipity elements, cubical meshes, cubes

[1] Alnæs, M.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M. E; Wells, G. N The FEniCS project version 1.5, Archive of Numerical Software, Volume 3 (2015) no. 100, pp. 9-23

[2] Alnæs, M.; Logg, A.; Ølgaard, K.; Rognes, M.; Wells, G. Unified form language: a domain-specific language for weak formulations of partial differential equations, ACM Transactions on Mathematical Software, Volume 40 (2014) no. 2, pp. 9:1-37 | DOI | MR | Zbl

[3] Arnold, D.; Awanou, G. Finite element differential forms on cubical meshes, Mathematics of Computation, Volume 83 (2014) no. 288, pp. 1551-1570 | DOI | MR | Zbl

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

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

[6] Arnold, D. N; Falk, R. S; Winther, R. Geometric decompositions and local bases for spaces of finite element differential forms, Computer Methods in Applied Mechanics and Engineering, Volume 198 (2009) no. 21-26, pp. 1660-1672 | DOI | MR | Zbl

[7] Arnold, D. N; Falk, R. S; Winther, R. 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 | Zbl

[8] Bangerth, W.; Hartmann, R.; Kanschat, G. deal.II—a general-purpose object-oriented finite element library, ACM Transactions on Mathematical Software (TOMS), Volume 33 (2007) no. 4, p. 24-es | DOI | MR | Zbl

[9] Bossavit, A. A uniform rationale for Whitney forms on various supporting shapes, Mathematics and Computers in Simulation, Volume 80 (2010) no. 8, pp. 1567-1577 | DOI | MR | Zbl

[10] Brezzi, F.; Douglas Jr, J.; Marini, L. D. Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, Volume 47 (1985) no. 2, pp. 217-235 | DOI | MR | Zbl

[11] Chen, W.; Wang, Y. Minimal degree $H$(curl) and $H$(div) conforming finite elements on polytopal meshes, Mathematics of Computation (2016) | DOI | Zbl

[12] Douglas, A.; Richard, F.; Ragnar, W. Finite element exterior calculus, homological techniques, and applications, Acta Numerica (2006), pp. 1-155 | MR | Zbl

[13] Gillette, A.; Kloefkorn, T. Trimmed serendipity finite element differential forms, Mathematics of Computation, Volume to appear (2018) | Zbl

[14] Gillette, A.; Rand, A.; Bajaj, C. Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes, Computational Methods in Applied Mathematics, Volume 16 (2016) no. 4, pp. 667-683 | DOI | MR | Zbl

[15] Nédélec, J.-C. Mixed finite elements in ${\mathbf{R}}^3$, Numerische Mathematik, Volume 35 (1980) no. 3, pp. 315-341 | DOI

[16] Nédélec, J.-C. A new family of mixed finite elements in ${\mathbf{R}}^3$, Numerische Mathematik, Volume 50 (1986) no. 1, pp. 57-81 | MR | Zbl

[17] Rathgeber, F.; Ham, D. A; Mitchell, L.; Lange, M.; Luporini, F.; McRae, A. TT; Bercea, G.-T.; Markall, G. R; Kelly, P. HJ Firedrake: automating the finite element method by composing abstractions, ACM Transactions on Mathematical Software (TOMS), Volume 43 (2017) no. 3, 24 pages | DOI | MR | Zbl

[18] Raviart, P.-A.; Thomas, J.-M. A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods, Springer, 1977, pp. 292-315 | DOI | Zbl

[19] The Sage Developers SageMath, the Sage Mathematics Software System (Version 7.6) (2017) (http://www.sagemath.org)

[20] Zaglmayr, S. High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität (2006) (Ph. D. Thesis)