An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow
HTML articles powered by AMS MathViewer
- by Daniele A. Di Pietro and Simon Lemaire PDF
- Math. Comp. 84 (2015), 1-31 Request permission
Abstract:
In this work we introduce a discrete functional space on general polygonal or polyhedral meshes which mimics two important properties of the standard Crouzeix–Raviart space, namely the continuity of mean values at interfaces and the existence of an interpolator which preserves the mean value of the gradient inside each element. The construction borrows ideas from both Cell Centered Galerkin and Hybrid Finite Volume methods. The discrete function space is defined from cell and face unknowns by introducing a suitable piecewise affine reconstruction on a (fictitious) pyramidal subdivision of the original mesh. Two applications are considered in which the discrete space plays an important role, namely \inparaenum[(i)]
the design of a locking-free primal (as opposed to mixed) method for quasi-incompressible planar elasticity on general polygonal meshes;
the design of an inf-sup stable method for the Stokes equations on general polygonal or polyhedral meshes. In this context, we also propose a general modification, applicable to any suitable discretization, which guarantees that the velocity approximation is unaffected by the presence of large irrotational body forces provided a Helmholtz decomposition of the right-hand side is available. \endinparaenum The relation between the proposed methods and classical finite volume and finite element schemes on standard meshes is investigated. Finally, similar ideas are exploited to mimic key properties of the lowest-order Raviart–Thomas space on general polygonal or polyhedral meshes.
References
- G. Allaire, Analyse Numérique et Optimisation, Les Éditions de l’École Polytechnique, Palaiseau, 2009.
- Douglas N. Arnold, Franco Brezzi, and Jim Douglas Jr., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347–367. MR 840802, DOI 10.1007/BF03167064
- Lourenco Beirão Da Veiga, A mimetic discretization method for linear elasticity, M2AN Math. Model. Numer. Anal. 44 (2010), no. 2, 231–250. MR 2655949, DOI 10.1051/m2an/2010001
- L. Beirão da Veiga, F. Brezzi, and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51 (2013), no. 2, 794–812. MR 3033033, DOI 10.1137/120874746
- L. Beirão da Veiga, V. Gyrya, K. Lipnikov, and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys. 228 (2009), no. 19, 7215–7232. MR 2568590, DOI 10.1016/j.jcp.2009.06.034
- L. Beirão da Veiga, K. Lipnikov, and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal. 48 (2010), no. 4, 1419–1443. MR 2684341, DOI 10.1137/090757411
- Susanne C. Brenner, Korn’s inequalities for piecewise $H^1$ vector fields, Math. Comp. 73 (2004), no. 247, 1067–1087. MR 2047078, DOI 10.1090/S0025-5718-03-01579-5
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
- Susanne C. Brenner and Li-Yeng Sung, Linear finite element methods for planar linear elasticity, Math. Comp. 59 (1992), no. 200, 321–338. MR 1140646, DOI 10.1090/S0025-5718-1992-1140646-2
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Franco Brezzi, Konstantin Lipnikov, and Mikhail Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43 (2005), no. 5, 1872–1896. MR 2192322, DOI 10.1137/040613950
- Franco Brezzi, Konstantin Lipnikov, Mikhail Shashkov, and Valeria Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3682–3692. MR 2339994, DOI 10.1016/j.cma.2006.10.028
- Franco Brezzi, Konstantin Lipnikov, and Valeria Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci. 15 (2005), no. 10, 1533–1551. MR 2168945, DOI 10.1142/S0218202505000832
- K. S. Chavan, B. P. Lamichhane, and B. I. Wohlmuth, Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D, Comput. Methods Appl. Mech. Engrg., 196:4075–4086, 2007.
- M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
- Daniele A. Di Pietro, Cell centered Galerkin methods for diffusive problems, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 111–144. MR 2846369, DOI 10.1051/m2an/2011016
- Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
- Daniele A. Di Pietro, Jean-Marc Gratien, and Christophe Prud’homme, A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes, BIT 53 (2013), no. 1, 111–152. MR 3029297, DOI 10.1007/s10543-012-0403-3
- Daniele A. Di Pietro and Serge Nicaise, A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media, Appl. Numer. Math. 63 (2013), 105–116. MR 2997905, DOI 10.1016/j.apnum.2012.09.009
- Jérôme Droniou and Robert Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math. 105 (2006), no. 1, 35–71. MR 2257385, DOI 10.1007/s00211-006-0034-1
- Jérôme Droniou and Robert Eymard, Study of the mixed finite volume method for Stokes and Navier-Stokes equations, Numer. Methods Partial Differential Equations 25 (2009), no. 1, 137–171. MR 2473683, DOI 10.1002/num.20333
- Jérôme Droniou, Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci. 20 (2010), no. 2, 265–295. MR 2649153, DOI 10.1142/S0218202510004222
- Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI 10.1090/S0025-5718-1980-0559195-7
- Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
- R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30 (2010), no. 4, 1009–1043. MR 2727814, DOI 10.1093/imanum/drn084
- Richard S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), no. 196, 529–550. MR 1094947, DOI 10.1090/S0025-5718-1991-1094947-6
- Keith J. Galvin, Alexander Linke, Leo G. Rebholz, and Nicholas E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg. 237/240 (2012), 166–176. MR 2947652, DOI 10.1016/j.cma.2012.05.008
- Peter Hansbo and Mats G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 17-18, 1895–1908. MR 1886000, DOI 10.1016/S0045-7825(01)00358-9
- Peter Hansbo and Mats G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, M2AN Math. Model. Numer. Anal. 37 (2003), no. 1, 63–72. MR 1972650, DOI 10.1051/m2an:2003020
- Bishnu P. Lamichhane and Ernst P. Stephan, A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems, Numer. Methods Partial Differential Equations 28 (2012), no. 4, 1336–1353. MR 2914794, DOI 10.1002/num.20683
- Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
- Gilbert Strang, Variational crimes in the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 689–710. MR 0413554
- Michael Vogelius, An analysis of the $p$-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates, Numer. Math. 41 (1983), no. 1, 39–53. MR 696549, DOI 10.1007/BF01396304
- Martin Vohralík and Barbara I. Wohlmuth, From face to element unknowns by local static condensation with application to nonconforming finite elements, Comput. Methods Appl. Mech. Engrg. 253 (2013), 517–529. MR 3002809, DOI 10.1016/j.cma.2012.08.013
- Martin Vohralík and Barbara I. Wohlmuth, Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods, Math. Models Methods Appl. Sci. 23 (2013), no. 5, 803–838. MR 3028542, DOI 10.1142/S0218202512500613
Additional Information
- Daniele A. Di Pietro
- Affiliation: Université Montpellier 2, I3M, 34057 Montpellier CEDEX 5, France
- Email: daniele.di-pietro@univ-montp2.fr
- Simon Lemaire
- Affiliation: IFP Énergies nouvelles, Department of Applied Mathematics, 1 & 4 avenue de Bois-Préau, 92852 Rueil-Malmaison CEDEX, France
- Email: simon.lemaire87@gmail.com
- Received by editor(s): November 19, 2012
- Received by editor(s) in revised form: June 5, 2013
- Published electronically: August 4, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 1-31
- MSC (2010): Primary 65N08, 65N30; Secondary 74B05, 76D07
- DOI: https://doi.org/10.1090/S0025-5718-2014-02861-5
- MathSciNet review: 3266951