Abstract
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of finite element systems, and the examples include conforming mixed finite elements for Stokes’ equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.
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Alfeld, P.: A trivariate Clough-Tocher scheme for tetrahedral data. Comput. Aided Geom. Des. 1(2), 169–181 (1984)
Alfeld, P., Sorokina, T.: Linear differential operators on bivariate spline spaces and spline vector fields. BIT 56(1), 15–32 (2016)
Arnold, D.N., Douglas Jr., J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45(1), 1–22 (1984)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)
Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure stokes elements. Adv. Comput. Methods Partial Differ. Equ. 7, 28–34 (1992)
Bernardi, C., Raugel, G.: Analysis of some finite elements for the Stokes problem. Math. Comput. 44(169), 71–79 (1985)
Brenner, S.C.: Forty years of the Crouzeix–Raviart element. Numer. Methods Partial Differ. Equ. 31(2), 367–396 (2015)
Christiansen, S.H.: Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numer. Math. 107(1), 87–106 (2007). [preprint at arXiv:1007.1120]
Christiansen, S.H.: A construction of spaces of compatible differential forms on cellular complexes. Math. Models Methods Appl. Sci. 18(5), 739–757 (2008)
Christiansen, S.H.: Foundations of Finite Element Methods for Wave Equations of Maxwell Type. Applied Wave Mathematics, pp. 335–393. Springer, Berlin (2009)
Christiansen, S.H., Gillette, A.: Constructions of some minimal finite element systems. Math. Model. Numer. Anal. 50(3), 833–850 (2016). [preprint at arXiv:1504.04670]
Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011)
Christiansen, S.H., Rapetti, F.: On high order finite element spaces of differential forms. Math. Comput. 85(296), 517–548 (2016). [preprint at arXiv:1306.4835]
Christiansen, S.H., Winther, R.: Smoothed projections in finite element exterior calculus. Math. Comput. 77(262), 813–829 (2008)
Ciarlet, P.G.: Sur l’élément de Clough et Tocher. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 19–27 (1974)
Ciarlet, P.G.: Basic Error Estimates for Elliptic Problems. Handbook of Numerical Analysis, vol. II, p. 17-351. North-Holland, Amsterdam (1991)
Demkowicz, L., Babuška, I.: \(p\) interpolation error estimates for edge finite elements of variable order in two dimensions. SIAM J. Numer. Anal. 41(4), 1195–1208 (2003). (electronic)
Demkowicz, L., Buffa, A.: \(H^1\), \(H({\rm curl})\) and \(H({\rm div})\)-conforming projection-based interpolation in three dimensions. Quasi-optimal \(p\)-interpolation estimates. Comput. Methods Appl. Mech. Eng. 194(2–5), 267–296 (2005)
Douglas, J., Dupont, T., Percell, P., Scott, R.: A family of \(C^{1}\) finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. 13(3), 227–255 (1979)
Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)
Godement, R.: Topologie algébrique et théorie des faisceaux. Hermann, Paris, 1973. Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, No. 1252 (1973)
Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34(4), 1489–1508 (2014)
Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014)
Hiptmair, R.: Canonical construction of finite elements. Math. Comput. 68(228), 1325–1346 (1999)
Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007)
Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)
Nédélec, J.-C.: Mixed finite elements in \({ R}^{3}\). Numer. Math. 35(3), 315–341 (1980)
Nédélec, J.-C.: Éléments finis mixtes incompressibles pour l’équation de Stokes dans \({ R}^{3}\). Numer. Math. 39(1), 97–112 (1982)
Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84(295), 2059–2081 (2015)
Percell, P.: On cubic and quartic Clough-Tocher finite elements. SIAM J. Numer. Anal. 13(1), 100–103 (1976)
Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids. Ph.D. Thesis, The Pennsylvania State University (1994)
Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods (Proceedings Conference on Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292–315. Lecture Notes in Mathematics, Vol. 606. Springer, Berlin (1977)
Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19(1), 111–143 (1985)
Stenberg, R.: Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comput. 42(165), 9–23 (1984)
Walkington, N.J.: A \(C^1\) tetrahedral finite element without edge degrees of freedom. SIAM J. Numer. Anal. 52(1), 330–342 (2014)
Worsey, A.J., Farin, G.: An \(n\)-dimensional Clough-Tocher interpolant. Constr. Approx. 3(2), 99–110 (1987)
Worsey, A.J., Piper, B.: A trivariate Powell–Sabin interpolant. Comput. Aided Geom. Des. 5(3), 177–186 (1988)
Zhang, S.: A new family of stable mixed finite elements for the 3d Stokes equations. Math. Comput. 74(250), 543–554 (2005)
Zhang, S.: On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26, 456–470 (2008)
Zhang, S.: Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids. Calcolo 48(3), 211–244 (2011)
Acknowledgements
We are grateful to Richard Falk for pointing out the paper [3], which has interesting connections with this one. We are also grateful to Shangyou Zhang for numerous bibliographical remarks. SHC is supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, Project 278011 STUCCOFIELDS. KH is supported by the China Scholarship Council (CSC), Project 201506010013 and by the European Research Council through the FP7-IDEAS-ERC Advanced Grant scheme, Project 650138 FEEC-A. The stimulating collaborations are achieved during his visit at University of Oslo (UiO) since September 2015. He is grateful for the kind hospitality and support of UiO.
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Christiansen, S.H., Hu, K. Generalized finite element systems for smooth differential forms and Stokes’ problem. Numer. Math. 140, 327–371 (2018). https://doi.org/10.1007/s00211-018-0970-6
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DOI: https://doi.org/10.1007/s00211-018-0970-6