Abstract
We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or how to derive the degrees of freedom.
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Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM J. Sci. Comput. 33(6), 3087–3109 (2011)
Ainsworth, M., Coyle, J.: Hierarchic finite element bases on unstructured tetrahedral meshes. International Journal for Numerical Methods in Engineering 58(14), 2103–2130 (2003)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)
Arnold, D.N., Falk, R.S., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198(21-26), 1660–1672 (2009)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: From Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)
Bentley, A.: Explicit construction of computational bases for RTk and BDMk spaces in R3. Computers & Mathematics with Applications 73(7), 1421–1432 (2017)
Beuchler, S., Pillwein, V., Zaglmayr, S.: Sparsity optimized high order finite element functions for h(div) on simplices. Numer. Math. 122(2), 197–225 (2012)
Beuchler, S., Pillwein, V., Zaglmayr, S.: Sparsity optimized high order finite element functions for h(curl) on tetrahedra. Adv. Appl. Math. 50(5), 749–769 (2013)
Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)
Christiansen, S.H., Rapetti, F.: On high order finite element spaces of differential forms. Mathematics of Computation. https://doi.org/10.1090/mcom/2995. Electronically published on July 10, 2015 (2015)
Ern, A., Guermond, J.L.: Finite elements i: Approximation and interpolation, vol. 72 Springer Nature (2021)
Ervin, V.: Computational bases for RTk and BDMk on triangles. Computers & Mathematics with Applications 64(8), 2765–2774 (2012)
Fuchs, D., Viro, O.: Topology II: Homotopy and homology. Classical manifolds springer. https://doi.org/10.1007/978-3-662-10581-8 (2004)
Gopalakrishnan, J., García-Castillo, L.E., Demkowicz, L.F.: Nédélec spaces in affine coordinates. Computers & Mathematics with Applications 49(7-8), 1285–1294 (2005)
Hiptmair, R.: Higher order Whitney forms. Progress in Electromagnetics Research 32, 271–299 (2001)
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11(1), 237–339 (2002)
Kirby, R.C.: Low-complexity finite element algorithms for the de Rham complex on simplices. SIAM J. Sci. Comput. 36(2), A846–A868 (2014)
Kirby, R. C.: Low-complexity finite element algorithms for the de Rham complex on simplices. SMAI Journal of Computational Mathematics 4, 197–224 (2018)
Lee, J. M.: Introduction to Smooth Manifolds Graduate Texts in Mathematics, 2nd edn., vol. 218. Springer, New York (2012)
Nédélec, J.C.: A new family of mixed finite elements in R3. Numer. Math. 50(1), 57–81 (1986)
Rapetti, F., Bossavit, A.: Geometrical localisation of the degrees of freedom for Whitney elements of higher order. Science, Measurement & Technology, IET 1(1), 63–66 (2007)
Rapetti, F., Bossavit, A.: Whitney forms of higher degree. SIAM J. Numer. Anal. 47, 2369–2386 (2009)
Schöberl, J., Zaglmayr, S.: High order nédélec elements with local complete sequence properties. COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 24 (2), 374–384 (2005)
Acknowledgements
The author acknowledges helpful discussions with Douglas N. Arnold and Snorre H. Christiansen. The referees’ careful reading and valuable comments are appreciated. The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Geometry, compatibility and structure preservation in computational differential equations” where work on this paper was undertaken.
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Open access funding provided by EPFL Lausanne. This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS. This research was supported in part by NSF DMS/RTG Award 1345013 and DMS/CM Award 1262982. This work was supported by EPSRC grant no EP/K032208/1.
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Communicated by: Francesca Rapetti
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Licht, M.W. On basis constructions in finite element exterior calculus. Adv Comput Math 48, 14 (2022). https://doi.org/10.1007/s10444-022-09926-6
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DOI: https://doi.org/10.1007/s10444-022-09926-6
Keywords
- Barycentric differential form
- Canonical spanning sets
- Degrees of freedom
- Finite element exterior calculus
- Geometrically decomposed bases