×

zbMATH — the first resource for mathematics

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. (English) Zbl 1201.65200
The authors study the unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. The authors give first some preliminaries and then give an abstract estimate on the energy norm of the difference between two arbitrary vector fields. Later this estimate is used to obtain both a priori and a posteriori estimates on the error approximation. Also they later mention about mixed finite element methods, postprocessing etc. Some complements on mixed finite element methods are given.

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 0951.65080 · doi:10.1137/S1064827595293582 · doi.org
[2] 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). · Zbl 0909.65076 · doi:10.1007/s002110050364 · doi.org
[3] 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. · Zbl 1050.76035 · doi:10.1007/s00211-002-0436-7 · doi.org
[4] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[5] Mark Ainsworth, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal. 42 (2005), no. 6, 2320 – 2341. · Zbl 1085.65102 · doi:10.1137/S0036142903425112 · doi.org
[6] 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. · Zbl 1159.65353 · doi:10.1137/06067331X · doi.org
[7] A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385 – 395. · Zbl 0866.65068 · doi:10.1007/s002110050222 · doi.org
[8] 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. · Zbl 0829.65127
[9] 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). · Zbl 0567.65078
[10] I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039 – 1062. · Zbl 0472.65083
[11] 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. · Zbl 0528.65046 · doi:10.1137/0720034 · doi.org
[12] Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322 – 333. · Zbl 0214.42001 · doi:10.1007/BF02165003 · doi.org
[13] M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751 – 756. · Zbl 1057.26011 · doi:10.4171/ZAA/1170 · doi.org
[14] 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). · Zbl 0962.65096 · doi:10.1007/PL00005393 · doi.org
[15] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431 – 2444. · Zbl 0866.65071 · doi:10.1137/S0036142994264079 · doi.org
[16] 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. · Zbl 0688.65061 · doi:10.1137/0726073 · doi.org
[17] 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 loose French summary). · Zbl 0338.90047
[18] 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. · Zbl 0631.65107 · doi:10.1007/BF01396752 · doi.org
[19] 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. · Zbl 0599.65072 · doi:10.1007/BF01389710 · doi.org
[20] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[21] 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. · Zbl 1108.65102 · doi:10.1137/040613950 · doi.org
[22] Erik Burman and Alexandre Ern, Continuous interior penalty \?\?-finite element methods for advection and advection-diffusion equations, Math. Comp. 76 (2007), no. 259, 1119 – 1140. · Zbl 1118.65118
[23] Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465 – 476. · Zbl 0864.65068
[24] 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. · Zbl 0997.65126
[25] 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). · Zbl 0784.65075
[26] 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. · Zbl 0932.65126
[27] 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. · Zbl 1007.65091 · doi:10.1137/S003614290037544X · doi.org
[28] 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. · Zbl 1084.65113 · doi:10.1137/S0036142902417893 · doi.org
[29] 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. · Zbl 1078.65093
[30] 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). · Zbl 0482.65057
[31] 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. · Zbl 0624.65109
[32] 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. · Zbl 1109.65099 · doi:10.1007/s00211-006-0034-1 · doi.org
[33] 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. · Zbl 0822.65086 · doi:10.1201/b16924-18 · doi.org
[34] 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. · Zbl 1077.65113 · doi:10.1051/m2an:2004044 · doi.org
[35] 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. · Zbl 1190.65165
[36] 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. · Zbl 0981.65095
[37] 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. · Zbl 1108.65099 · doi:10.1007/s00211-006-0036-z · doi.org
[38] 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. · Zbl 0801.65108
[39] 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. · Zbl 0889.65124 · doi:10.1137/S0036142994276992 · doi.org
[40] 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. · Zbl 1058.65120 · doi:10.1137/S0036142902405217 · doi.org
[41] Kwang Y. Kim, A posteriori error analysis for locally conservative mixed methods, Math. Comp. 76 (2007), no. 257, 43 – 66. · Zbl 1121.65112
[42] Robert Kirby, Residual a posteriori error estimates for the mixed finite element method, Comput. Geosci. 7 (2003), no. 3, 197 – 214. · Zbl 1033.65096 · doi:10.1023/A:1025518113877 · doi.org
[43] 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. · Zbl 1136.65101 · doi:10.1007/s00211-007-0121-y · doi.org
[44] Carlo Lovadina and Rolf Stenberg, Energy norm a posteriori error estimates for mixed finite element methods, Math. Comp. 75 (2006), no. 256, 1659 – 1674. · Zbl 1119.65110
[45] 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). · Zbl 0711.65091
[46] 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. · Zbl 0573.65082 · doi:10.1137/0722029 · doi.org
[47] J.-C. Nédélec, Mixed finite elements in \?³, Numer. Math. 35 (1980), no. 3, 315 – 341. · Zbl 0419.65069 · doi:10.1007/BF01396415 · doi.org
[48] 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. · Zbl 1112.65111
[49] L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286 – 292 (1960). · Zbl 0099.08402 · doi:10.1007/BF00252910 · doi.org
[50] 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. · Zbl 1267.65165
[51] W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241 – 269. · Zbl 0029.23505
[52] Alfio Quarteroni and Alberto Valli, Numerical approximation of partial differential equations, Springer Series in Computational Mathematics, vol. 23, Springer-Verlag, Berlin, 1994. · Zbl 0803.65088
[53] 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) Springer, Berlin, 1977, pp. 292 – 315. Lecture Notes in Math., Vol. 606.
[54] 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. · Zbl 1086.65103 · doi:10.1163/156939805775122271 · doi.org
[55] 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. · Zbl 1185.35048 · doi:10.1137/050641533 · doi.org
[56] 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. · Zbl 0875.65090
[57] Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151 – 167 (English, with French summary). · Zbl 0717.65081
[58] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Teubner-Wiley, Stuttgart, 1996. · Zbl 0853.65108
[59] Martin Vohralík, On the discrete Poincaré-Friedrichs inequalities for nonconforming approximations of the Sobolev space \?\textonesuperior , Numer. Funct. Anal. Optim. 26 (2005), no. 7-8, 925 – 952. · Zbl 1089.65124 · doi:10.1080/01630560500444533 · doi.org
[60] 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. · Zbl 1116.65121 · doi:10.1051/m2an:2006013 · doi.org
[61] 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). · Zbl 1359.76199 · doi:10.1051/proc:071806 · doi.org
[62] 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. · Zbl 1151.65084 · doi:10.1137/060653184 · doi.org
[63] -, 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
[64] 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. · Zbl 1160.65059 · doi:10.1007/s00211-008-0168-4 · doi.org
[65] 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. · Zbl 1112.65123
[66] 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. · Zbl 1094.65114 · doi:10.1137/S0036142903431687 · doi.org
[67] 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. · Zbl 0929.65094
[68] 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. · Zbl 1043.65131 · doi:10.1002/nme.874 · doi.org
[69] 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. · Zbl 0602.73063 · doi:10.1002/nme.1620240206 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.