Trimmed serendipity finite element differential forms

Gillette, Andrew; Kloefkorn, Tyler

doi:10.1090/mcom/3354

Trimmed serendipity finite element differential forms
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by Andrew Gillette and Tyler Kloefkorn;

Math. Comp. 88 (2019), 583-606

DOI: https://doi.org/10.1090/mcom/3354

Published electronically: May 18, 2018

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Abstract:

We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83 (2014), pp. 1551–1570] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order $r$ provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50 (2016), pp. 883–850], namely, a minimal compatible finite element system on squares or cubes containing order $r-1$ polynomial differential forms.

References

R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687, DOI 10.1007/978-1-4612-1029-0
Todd Arbogast and Maicon R. Correa, Two families of $H(\textrm {div})$ mixed finite elements on quadrilaterals of minimal dimension, SIAM J. Numer. Anal. 54 (2016), no. 6, 3332–3356. MR 3574588, DOI 10.1137/15M1013705
Todd Arbogast, Gergina Pencheva, Mary F. Wheeler, and Ivan Yotov, A multiscale mortar mixed finite element method, Multiscale Model. Simul. 6 (2007), no. 1, 319–346. MR 2306414, DOI 10.1137/060662587
Douglas N. Arnold and Gerard Awanou, The serendipity family of finite elements, Found. Comput. Math. 11 (2011), no. 3, 337–344. MR 2794906, DOI 10.1007/s10208-011-9087-3
Douglas N. Arnold and Gerard Awanou, Finite element differential forms on cubical meshes, Math. Comp. 83 (2014), no. 288, 1551–1570. MR 3194121, DOI 10.1090/S0025-5718-2013-02783-4
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
D. Arnold and A. Logg, Periodic table of the finite elements, SIAM News, 47(9), 2014. femtable.org.
Douglas N. Arnold and Hongtao Chen, Finite element exterior calculus for parabolic problems, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 1, 17–34. MR 3600999, DOI 10.1051/m2an/2016013
L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, Serendipity nodal VEM spaces, Comput. & Fluids 141 (2016), 2–12. MR 3569211, DOI 10.1016/j.compfluid.2016.02.015
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Snorre H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739–757. MR 2413036, DOI 10.1142/S021820250800284X
Snorre H. Christiansen, Foundations of finite element methods for wave equations of Maxwell type, Applied wave mathematics, Springer, Berlin, 2009, pp. 335–393. MR 2553283, DOI 10.1007/978-3-642-00585-5_{1}7
Snorre H. Christiansen and Andrew Gillette, Constructions of some minimal finite element systems, ESAIM Math. Model. Numer. Anal. 50 (2016), no. 3, 833–850. MR 3507275, DOI 10.1051/m2an/2015089
Snorre H. Christiansen, Hans Z. Munthe-Kaas, and Brynjulf Owren, Topics in structure-preserving discretization, Acta Numer. 20 (2011), 1–119. MR 2805152, DOI 10.1017/S096249291100002X
Snorre H. Christiansen and Francesca Rapetti, On high order finite element spaces of differential forms, Math. Comp. 85 (2016), no. 298, 517–548. MR 3434870, DOI 10.1090/mcom/2995
Bernardo Cockburn and Guosheng Fu, A systematic construction of finite element commuting exact sequences, SIAM J. Numer. Anal. 55 (2017), no. 4, 1650–1688. MR 3670721, DOI 10.1137/16M1073352
Andrew Gillette, Michael Holst, and Yunrong Zhu, Finite element exterior calculus for evolution problems, J. Comput. Math. 35 (2017), no. 2, 187–212. MR 3623353, DOI 10.4208/jcm.1610-m2015-0319
R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, DOI 10.1090/S0025-5718-99-01166-7
Anil Nirmal Hirani, Discrete exterior calculus, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–California Institute of Technology. MR 2704508
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 292–315. MR 483555