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Analysis and Numerics of Partial Differential Equations pp 117–140Cite as

Spaces of Finite Element Differential Forms

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 4))

Abstract

We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit commuting projections. We present two families of spaces in the case of simplicial meshes, and two other families in the case of cubical meshes. We make use of the exterior calculus and the Koszul complex to define and understand the spaces. These tools allow us to treat a wide variety of situations, which are often treated separately, in a unified fashion.

The work of the author was supported by NSF grant DMS-1115291.

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References

  1. Arnold, D.N.: Differential complexes and numerical stability. In: Proceedings of the International Congress of Mathematicians, vol. I, Beijing, 2002, pp. 137–157. Higher Education Press, Beijing (2002). MR MR1989182 (2004h:65115)

    Google Scholar 

  2. Arnold, D.N., Awanou, G.: The serendipity family of finite elements. Found. Comput. Math. 11(3), 337–344 (2011). doi:10.1007/s10208-011-9087-3

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Awanou, G.: Finite element differential forms on cubical meshes. Preprint (2012). URL: http://arxiv.org/pdf/1204.2595

  4. Arnold, D.N., Boffi, D., Bonizzoni, F.: Approximation by tensor product finite element differential forms (2012, in preparation)

    Google Scholar 

  5. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006). MR MR2269741 (2007j:58002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 42(2), 281–354 (2010)

    Article  MathSciNet  Google Scholar 

  7. Baker, G.A.: Combinatorial Laplacians and Sullivan-Whitney forms. In: Differential Geometry, College Park, MD, 1981/1982. Progr. Math., vol. 32, pp. 1–33. Birkhäuser, Boston (1983). MR MR702525 (84m:58005)

    Google Scholar 

  8. Bossavit, A.: Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEEE Trans. Magn. 135(Part A), 493–500 (1988)

    Google Scholar 

  9. Brezzi, F., Douglas, J. Jr., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987). MR MR890035 (88f:65190)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985). MR MR799685 (87g:65133)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). MR MR0520174 (58 #25001)

    MATH  Google Scholar 

  12. Cockburn, B., Qiu, W.: Commuting diagrams for the TNT elements on cubes. Math. Comput. (2012, to appear)

    Google Scholar 

  13. Dodziuk, J.: Finite-difference approach to the Hodge theory of harmonic forms. Am. J. Math. 98(1), 79–104 (1976). MR MR0407872 (53 #11642)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dodziuk, J., Patodi, V.K.: Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. (N.S.) 40(1–4), 1–52 (1976). MR MR0488179 (58 #7742)

    MathSciNet  MATH  Google Scholar 

  15. Hiptmair, R.: Canonical construction of finite elements. Math. Comput. 68, 1325–1346 (1999). MR MR1665954 (2000b:65214)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kotiuga, P.R.: Hodge decompositions and computational electromagnetics. PhD in Electrical Engineering, McGill University (1984)

    Google Scholar 

  17. Müller, W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28(3), 233–305 (1978). MR MR498252 (80j:58065b)

    Article  MATH  Google Scholar 

  18. Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Math. 35, 315–341 (1980). MR MR592160 (81k:65125)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nédélec, J.-C.: A new family of mixed finite elements in R 3. Numer. Math. 50, 57–81 (1986). MR MR864305 (88e:65145)

    Article  MathSciNet  MATH  Google Scholar 

  20. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975, pp. 292–315. Lecture Notes in Mathematics, vol. 606. Springer, Berlin (1977). MR MR0483555 (58 #3547)

    Chapter  Google Scholar 

  21. Robin, J.W., Salamon, D.A.: Introduction to differential topology (2011). Lecture notes for a course at ETH Zürich. URL: http://www.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf

  22. Sullivan, D.: Differential forms and the topology of manifolds. In: Manifolds—Tokyo 1973, Proc. Internat. Conf., Tokyo, 1973, pp. 37–49. Univ. Tokyo Press, Tokyo (1975). MR MR0370611 (51 #6838)

    Google Scholar 

  23. Sullivan, D.: Infinitesimal computations in topology. Publ. Math. IHÉS 1977(47), 269–331 (1978). MR MR0646078 (58 #31119)

    Google Scholar 

  24. Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957). MR MR0087148 (19,309c)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Douglas N. Arnold

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  1. Douglas N. Arnold
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Correspondence to Douglas N. Arnold .

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Editors and Affiliations

  1. Ist. Matematica Applicata e, Tecnologie Informatiche (IMATI), CNR Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Franco Brezzi

  2. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Piero Colli Franzone

  3. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, Pavia, 27100, Italy

    Ugo Gianazza

  4. Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, via Ferrata 1, Pavia, 27100, Pavia, Italy

    Gianni Gilardi

Additional information

In memory of Enrico Magenes, in gratitude for his deep and elegant mathematics, which taught us, and his profound humanity, which inspired us.

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Arnold, D.N. (2013). Spaces of Finite Element Differential Forms. In: Brezzi, F., Colli Franzone, P., Gianazza, U., Gilardi, G. (eds) Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol 4. Springer, Milano. https://doi.org/10.1007/978-88-470-2592-9_9

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