Constructions of some minimal finite element systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 833-850.

Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these constraints. We show how various known mixed finite element spaces fulfill such a design principle, including trimmed polynomial differential forms, serendipity elements and TNT elements. We also comment on virtual element methods and provide a dimension formula for minimal compatible finite element systems containing polynomials of a given degree on hypercubes.

Reçu le :
DOI : 10.1051/m2an/2015089
Classification : 65N30
Mots clés : finite element systems, differential forms, virtual element methods, Serendipity elements, TNT elements
Christiansen, Snorre H. 1 ; Gillette, Andrew 2

1 Department of Mathematics, University of Oslo, PO Box 1053 Blindern, 0316 Oslo, Norway
2 Department of Mathematics, University of Arizona, PO Box 210089, Tucson, Arizona, USA
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Christiansen, Snorre H.; Gillette, Andrew. Constructions of some minimal finite element systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 833-850. doi : 10.1051/m2an/2015089. http://archive.numdam.org/articles/10.1051/m2an/2015089/

D.N. Arnold and G. Awanou, Finite element differential forms on cubical meshes. Math. Comput. 83 (2014) 1551–1570. | DOI | MR | Zbl

D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. | DOI | MR | Zbl

D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.) 47 (2010) 281–354. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming VEM. Preprint (2014). | arXiv

F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. | DOI | MR | Zbl

S.H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numer. Math. 107 (2007) 87–106. | DOI | MR | Zbl

S.H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes. Math. Models Methods Appl. Sci. 18 (2008) 739–757. | DOI | MR | Zbl

S.H. Christiansen, Foundations of Finite Element Methods for Wave Equations of Maxwell Type. In Applied Wave Mathematics. Springer, Berlin, Heidelberg (2009) 335–393. | MR | Zbl

S.H. Christiansen, Éléments finis mixtes minimaux sur les polyèdres. C. R. Math. Acad. Sci. Paris 348 (2010) 217–221. | DOI | MR | Zbl

S.H. Christiansen, Upwinding in Finite Element Systems of Differential Forms. In Foundations of Computational Mathematics, Budapest 2011. Vol. 403 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge (2013) 45–71. | MR

S.H. Christiansen and F. Rapetti, On high order finite element spaces of differential forms. Math. Comput. 85 (2015) 517–547. | DOI | MR

S.H. Christiansen, H.Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization. Acta Numer. 20 (2011) 1–119. | DOI | MR | Zbl

S.H. Christiansen, T.G. Halvorsen and T.M. Sørensen, Stability of an upwind petrov galerkin discretization of convection diffusion equations. Preprint (2014). | arXiv

B. Cockburn and W. Qiu, Commuting diagrams for the TNT elements on cubes. Math. Comput. 83 (2014) 603–633. | DOI | MR | Zbl

R. Hiptmair, Canonical construction of finite elements. Math. Comput. 68 (1999) 1325–1346. | DOI | MR | Zbl

J.-C. Nédélec, Mixed finite elements in 𝐑 3 . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

P.-A. Raviart and J.M. Thomas, 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. Vol. 606 of Lect. Notes Math. Springer, Berlin (1977) 292–315. | MR | Zbl

M.E. Taylor, Partial Differential Equations I: Basic Theory. Vol. 115 of Appl. Math. Sci. Springer-Verlag, New York (1996). | MR | Zbl

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