Finite element differential forms on cubical meshes
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- by Douglas N. Arnold and Gerard Awanou PDF
- Math. Comp. 83 (2014), 1551-1570 Request permission
Abstract:
We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new $H(\mathrm {curl})$ and $H(\mathrm {div})$ finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.References
- Scot Adams and Victor Reiner, private communication.
- 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, Daniele Boffi, and Francesca Bonizzoni, Tensor product finite element differential forms and their approximation properties, preprint 2012, arXiv: 1212.6559 [math.NA].
- 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
- Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
- 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
- Runchang Lin and Zhimin Zhang, Natural superconvergence points in three-dimensional finite elements, SIAM J. Numer. Anal. 46 (2008), no. 3, 1281–1297. MR 2390994, DOI 10.1137/070681168
- Jean-Claude Nédélec, Mixed finite elements in $\textbf {R}^ {3}$, Numerische Mathematik 35 (1980), 315–241.
- 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, 1977, pp. 292–315. MR 0483555
- Barna Szabó and Ivo Babuška, Finite element analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. MR 1164869
Additional Information
- Douglas N. Arnold
- Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 27240
- Email: arnold@umn.edu
- Gerard Awanou
- Affiliation: Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois at Chicago, Chicago, Illinois 60607-7045
- MR Author ID: 700956
- Email: awanou@uic.edu
- Received by editor(s): April 11, 2012
- Received by editor(s) in revised form: December 21, 2012
- Published electronically: October 17, 2013
- Additional Notes: The work of the first author was supported in part by NSF grant DMS-1115291.
The work of the second author was supported in part by NSF grant DMS-0811052 and the Sloan Foundation. - © Copyright 2013 American Mathematical Society
- Journal: Math. Comp. 83 (2014), 1551-1570
- MSC (2010): Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2013-02783-4
- MathSciNet review: 3194121