Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods
HTML articles powered by AMS MathViewer
- by Martin Vohralík PDF
- Math. Comp. 79 (2010), 2001-2032 Request permission
Abstract:
We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), 943–972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete $\inf$–$\sup$ condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion–dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the $H^1_0(\Omega )$ space and also establish several links between mixed finite element approximations and some generalized weak solutions.References
- I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods, SIAM J. Sci. Comput. 19 (1998), no. 5, 1700–1716. MR 1618761, DOI 10.1137/S1064827595293582
- B. Achchab, A. Agouzal, J. Baranger, and J. F. Maitre, Estimateur d’erreur a posteriori hiérarchique. Application aux éléments finis mixtes, Numer. Math. 80 (1998), no. 2, 159–179 (French, with English and French summaries). MR 1645037, DOI 10.1007/s002110050364
- Y. Achdou, C. Bernardi, and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations, Numer. Math. 96 (2003), no. 1, 17–42. MR 2018789, DOI 10.1007/s00211-002-0436-7
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Mark Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal. 42 (2005), no. 6, 2320–2341. MR 2139395, DOI 10.1137/S0036142903425112
- Mark Ainsworth, A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements, SIAM J. Sci. Comput. 30 (2007/08), no. 1, 189–204. MR 2377438, DOI 10.1137/06067331X
- A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385–395. MR 1414415, DOI 10.1007/s002110050222
- Todd Arbogast and Zhangxin Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), no. 211, 943–972. MR 1303084, DOI 10.1090/S0025-5718-1995-1303084-8
- D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
- I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039–1062. MR 583486, DOI 10.1090/S0025-5718-1980-0583486-7
- I. Babuška and J. E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20 (1983), no. 3, 510–536. MR 701094, DOI 10.1137/0720034
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751–756. MR 2036927, DOI 10.4171/ZAA/1170
- C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math. 85 (2000), no. 4, 579–608 (English, with English and French summaries). MR 1771781, DOI 10.1007/PL00005393
- D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431–2444. MR 1427472, DOI 10.1137/S0036142994264079
- James H. Bramble and Jinchao Xu, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal. 26 (1989), no. 6, 1267–1275. MR 1025087, DOI 10.1137/0726073
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
- Franco Brezzi, Jim Douglas Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. MR 890035, DOI 10.1007/BF01396752
- Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
- 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
- Erik Burman and Alexandre Ern, Continuous interior penalty $hp$-finite element methods for advection and advection-diffusion equations, Math. Comp. 76 (2007), no. 259, 1119–1140. MR 2299768, DOI 10.1090/S0025-5718-07-01951-5
- Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465–476. MR 1408371, DOI 10.1090/S0025-5718-97-00837-5
- Carsten Carstensen and Sören Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945–969. MR 1898741, DOI 10.1090/S0025-5718-02-01402-3
- Zhangxin Chen, Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 1, 9–34 (English, with English and French summaries). MR 1204626, DOI 10.1051/m2an/1993270100091
- Zhangxin Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems, East-West J. Numer. Math. 4 (1996), no. 1, 1–33. MR 1393063
- So-Hsiang Chou, Do Y. Kwak, and Kwang Y. Kim, A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: the overlapping covolume case, SIAM J. Numer. Anal. 39 (2001), no. 4, 1170–1196. MR 1870838, DOI 10.1137/S003614290037544X
- Bernardo Cockburn and Jayadeep Gopalakrishnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 1, 283–301. MR 2051067, DOI 10.1137/S0036142902417893
- Bernardo Cockburn and Jayadeep Gopalakrishnan, Error analysis of variable degree mixed methods for elliptic problems via hybridization, Math. Comp. 74 (2005), no. 252, 1653–1677. MR 2164091, DOI 10.1090/S0025-5718-05-01741-2
- J. Douglas Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103 (English, with Portuguese summary). MR 667620
- Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39–52. MR 771029, DOI 10.1090/S0025-5718-1985-0771029-9
- 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
- Ricardo Durán and Claudio Padra, An error estimator for nonconforming approximations of a nonlinear problem, Finite element methods (Jyväskylä, 1993) Lecture Notes in Pure and Appl. Math., vol. 164, Dekker, New York, 1994, pp. 201–205. MR 1299990, DOI 10.1201/b16924-18
- Linda El Alaoui and Alexandre Ern, Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods, M2AN Math. Model. Numer. Anal. 38 (2004), no. 6, 903–929. MR 2108938, DOI 10.1051/m2an:2004044
- Alexandre Ern, Annette F. Stephansen, and Martin Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection–diffusion–reaction problems, J. Comput. Appl. Math. 234 (2010), no. 1, 114–130.
- Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
- Robert Eymard, Danielle Hilhorst, and Martin Vohralík, A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numer. Math. 105 (2006), no. 1, 73–131. MR 2257386, DOI 10.1007/s00211-006-0036-z
- Richard S. Falk and John E. Osborn, Remarks on mixed finite element methods for problems with rough coefficients, Math. Comp. 62 (1994), no. 205, 1–19. MR 1203735, DOI 10.1090/S0025-5718-1994-1203735-1
- Ronald H. W. Hoppe and Barbara Wohlmuth, Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems, SIAM J. Numer. Anal. 34 (1997), no. 4, 1658–1681. MR 1461801, DOI 10.1137/S0036142994276992
- Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, DOI 10.1137/S0036142902405217
- Kwang Y. Kim, A posteriori error analysis for locally conservative mixed methods, Math. Comp. 76 (2007), no. 257, 43–66. MR 2261011, DOI 10.1090/S0025-5718-06-01903-X
- Robert Kirby, Residual a posteriori error estimates for the mixed finite element method, Comput. Geosci. 7 (2003), no. 3, 197–214. MR 2007578, DOI 10.1023/A:1025518113877
- Mats G. Larson and Axel Målqvist, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math. 108 (2008), no. 3, 487–500. MR 2365826, DOI 10.1007/s00211-007-0121-y
- Carlo Lovadina and Rolf Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75 (2006), no. 256, 1659–1674. MR 2240629, DOI 10.1090/S0025-5718-06-01872-2
- L. Donatella Marini and P. Pietra, An abstract theory for mixed approximations of second order elliptic problems, Mat. Apl. Comput. 8 (1989), no. 3, 219–239 (English, with Portuguese summary). MR 1067287
- Luisa Donatella Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method, SIAM J. Numer. Anal. 22 (1985), no. 3, 493–496. MR 787572, DOI 10.1137/0722029
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- Serge Nicaise and Emmanuel Creusé, Isotropic and anisotropic a posteriori error estimation of the mixed finite element method for second order operators in divergence form, Electron. Trans. Numer. Anal. 23 (2006), 38–62. MR 2268551
- L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
- Gergina V. Pencheva, Martin Vohralík, Mary F. Wheeler, and Tim Wildey, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling, Preprint R10015, Laboratoire Jacques-Louis Lions, HAL Preprint 00467738, Submitted for publication, 2010.
- W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
- Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. MR 1299729, DOI 10.1007/978-3-540-85268-1
- P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
- S. I. Repin and A. Smolianski, Functional-type a posteriori error estimates for mixed finite element methods, Russian J. Numer. Anal. Math. Modelling 20 (2005), no. 4, 365–382. MR 2161108, DOI 10.1163/156939805775122271
- Sergey Repin, Stefan Sauter, and Anton Smolianski, Two-sided a posteriori error estimates for mixed formulations of elliptic problems, SIAM J. Numer. Anal. 45 (2007), no. 3, 928–945. MR 2318795, DOI 10.1137/050641533
- J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 523–639. MR 1115239
- Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511
- R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Teubner-Wiley, Stuttgart, 1996.
- Martin Vohralík, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space $H^1$, Numer. Funct. Anal. Optim. 26 (2005), no. 7-8, 925–952. MR 2192029, DOI 10.1080/01630560500444533
- Martin Vohralík, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes, M2AN Math. Model. Numer. Anal. 40 (2006), no. 2, 367–391. MR 2241828, DOI 10.1051/m2an:2006013
- Martin Vohralík, A posteriori error estimates for finite volume and mixed finite element discretizations of convection-diffusion-reaction equations, Paris-Sud Working Group on Modelling and Scientific Computing 2006–2007, ESAIM Proc., vol. 18, EDP Sci., Les Ulis, 2007, pp. 57–69 (English, with English and French summaries). MR 2404896, DOI 10.1051/proc:071806
- Martin Vohralík, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal. 45 (2007), no. 4, 1570–1599. MR 2338400, DOI 10.1137/060653184
- —, Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, HAL Preprint 00235810, version 2, submitted for publication, 2009
- Martin Vohralík, Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods, Numer. Math. 111 (2008), no. 1, 121–158. MR 2448206, DOI 10.1007/s00211-008-0168-4
- Martin Vohralík, Jiří Maryška, and Otto Severýn, Mixed and nonconforming finite element methods on a system of polygons, Appl. Numer. Math. 57 (2007), 176–193.
- Mary F. Wheeler and Ivan Yotov, A posteriori error estimates for the mortar mixed finite element method, SIAM J. Numer. Anal. 43 (2005), no. 3, 1021–1042. MR 2177794, DOI 10.1137/S0036142903431687
- Barbara I. Wohlmuth and Ronald H. W. Hoppe, A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements, Math. Comp. 68 (1999), no. 228, 1347–1378. MR 1651760, DOI 10.1090/S0025-5718-99-01125-4
- Anis Younes, Philippe Ackerer, and Guy Chavent, From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions, Internat. J. Numer. Methods Engrg. 59 (2004), no. 3, 365–388. MR 2029282, DOI 10.1002/nme.874
- O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337–357. MR 875306, DOI 10.1002/nme.1620240206
Additional Information
- Martin Vohralík
- Affiliation: UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France –and– CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France
- ORCID: 0000-0002-8838-7689
- Email: vohralik@ann.jussieu.fr
- Received by editor(s): July 7, 2008
- Received by editor(s) in revised form: August 6, 2009
- Published electronically: May 26, 2010
- Additional Notes: This work was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 2001-2032
- MSC (2010): Primary 65N15, 65N30, 76S05
- DOI: https://doi.org/10.1090/S0025-5718-2010-02375-0
- MathSciNet review: 2684353