{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T04:17:34Z","timestamp":1773807454945,"version":"3.50.1"},"reference-count":69,"publisher":"American Mathematical Society (AMS)","issue":"272","license":[{"start":{"date-parts":[[2011,5,26]],"date-time":"2011-05-26T00:00:00Z","timestamp":1306368000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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, <italic>Math. Comp.<\/italic>\u00a0<bold>64<\/bold> (1995), 943\u2013972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"inf\">\n  <mml:semantics>\n    <mml:mo movablelimits=\"true\" form=\"prefix\">inf<\/mml:mo>\n    <mml:annotation encoding=\"application\/x-tex\">\\inf<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>\u2013<inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sup\">\n  <mml:semantics>\n    <mml:mo movablelimits=\"true\" form=\"prefix\">sup<\/mml:mo>\n    <mml:annotation encoding=\"application\/x-tex\">\\sup<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> 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\u2013dispersion 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 <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H 0 Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msubsup>\n        <mml:mi>H<\/mml:mi>\n        <mml:mn>0<\/mml:mn>\n        <mml:mn>1<\/mml:mn>\n      <\/mml:msubsup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">H^1_0(\\Omega )<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> space and also establish several links between mixed finite element approximations and some generalized weak solutions.<\/p>","DOI":"10.1090\/s0025-5718-2010-02375-0","type":"journal-article","created":{"date-parts":[[2010,7,6]],"date-time":"2010-07-06T21:17:12Z","timestamp":1278451032000},"page":"2001-2032","source":"Crossref","is-referenced-by-count":57,"title":["Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods"],"prefix":"10.1090","volume":"79","author":[{"given":"Martin","family":"Vohral\u00edk","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,5,26]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1700","DOI":"10.1137\/S1064827595293582","article-title":"Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods","volume":"19","author":"Aavatsmark, I.","year":"1998","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s002110050364","article-title":"Estimateur d\u2019erreur a posteriori hi\u00e9rarchique. Application aux \u00e9l\u00e9ments finis mixtes","volume":"80","author":"Achchab, B.","year":"1998","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s00211-002-0436-7","article-title":"A priori and a posteriori analysis of finite volume discretizations of Darcy\u2019s equations","volume":"96","author":"Achdou, Y.","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"4","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2320","DOI":"10.1137\/S0036142903425112","article-title":"Robust a posteriori error estimation for nonconforming finite element approximation","volume":"42","author":"Ainsworth, Mark","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1137\/06067331X","article-title":"A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements","volume":"30","author":"Ainsworth, Mark","year":"2007","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s002110050222","article-title":"Error estimators for a mixed method","volume":"74","author":"Alonso, A.","year":"1996","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"211","key":"8","doi-asserted-by":"publisher","first-page":"943","DOI":"10.2307\/2153478","article-title":"On the implementation of mixed methods as nonconforming methods for second-order elliptic problems","volume":"64","author":"Arbogast, Todd","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1051\/m2an\/1985190100071","article-title":"Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates","volume":"19","author":"Arnold, D. N.","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"152","key":"10","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.2307\/2006374","article-title":"Analysis of mixed methods using mesh dependent norms","volume":"35","author":"Babu\u0161ka, I.","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1137\/0720034","article-title":"Generalized finite element methods: their performance and their relation to mixed methods","volume":"20","author":"Babu\u0161ka, I.","year":"1983","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"12","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF02165003","article-title":"Error-bounds for finite element method","volume":"16","author":"Babu\u0161ka, Ivo","year":"1970","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"13","doi-asserted-by":"publisher","first-page":"751","DOI":"10.4171\/ZAA\/1170","article-title":"A note on the Poincar\u00e9 inequality for convex domains","volume":"22","author":"Bebendorf, M.","year":"2003","journal-title":"Z. Anal. Anwendungen","ISSN":"https:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1007\/PL00005393","article-title":"Adaptive finite element methods for elliptic equations with non-smooth coefficients","volume":"85","author":"Bernardi, C.","year":"2000","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"6","key":"15","doi-asserted-by":"publisher","first-page":"2431","DOI":"10.1137\/S0036142994264079","article-title":"A posteriori error estimators for the Raviart-Thomas element","volume":"33","author":"Braess, D.","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"16","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1137\/0726073","article-title":"A local post-processing technique for improving the accuracy in mixed finite-element approximations","volume":"26","author":"Bramble, James H.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"17","first-page":"129","article-title":"On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers","volume":"8","author":"Brezzi, F.","year":"1974","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF01396752","article-title":"Mixed finite elements for second order elliptic problems in three variables","volume":"51","author":"Brezzi, Franco","year":"1987","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF01389710","article-title":"Two families of mixed finite elements for second order elliptic problems","volume":"47","author":"Brezzi, Franco","year":"1985","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"20","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"5","key":"21","doi-asserted-by":"publisher","first-page":"1872","DOI":"10.1137\/040613950","article-title":"Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes","volume":"43","author":"Brezzi, Franco","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"259","key":"22","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1090\/S0025-5718-07-01951-5","article-title":"Continuous interior penalty \u210e\ud835\udc5d-finite element methods for advection and advection-diffusion equations","volume":"76","author":"Burman, Erik","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"218","key":"23","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1090\/S0025-5718-97-00837-5","article-title":"A posteriori error estimate for the mixed finite element method","volume":"66","author":"Carstensen, Carsten","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"239","key":"24","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1090\/S0025-5718-02-01402-3","article-title":"Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM","volume":"71","author":"Carstensen, Carsten","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1051\/m2an\/1993270100091","article-title":"Analysis of mixed methods using conforming and nonconforming finite element methods","volume":"27","author":"Chen, Zhangxin","year":"1993","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"26","first-page":"1","article-title":"Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems","volume":"4","author":"Chen, Zhangxin","year":"1996","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"4","key":"27","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1137\/S003614290037544X","article-title":"A general framework for constructing and analyzing mixed finite volume methods on quadrilateral grids: the overlapping covolume case","volume":"39","author":"Chou, So-Hsiang","year":"2001","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1137\/S0036142902417893","article-title":"A characterization of hybridized mixed methods for second order elliptic problems","volume":"42","author":"Cockburn, Bernardo","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"252","key":"29","doi-asserted-by":"publisher","first-page":"1653","DOI":"10.1090\/S0025-5718-05-01741-2","article-title":"Error analysis of variable degree mixed methods for elliptic problems via hybridization","volume":"74","author":"Cockburn, Bernardo","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"30","first-page":"91","article-title":"Mixed finite element methods for second order elliptic problems","volume":"1","author":"Douglas, J., Jr.","year":"1982","journal-title":"Mat. Apl. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0101-8205","issn-type":"print"},{"issue":"169","key":"31","doi-asserted-by":"publisher","first-page":"39","DOI":"10.2307\/2007791","article-title":"Global estimates for mixed methods for second order elliptic equations","volume":"44","author":"Douglas, Jim, Jr.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s00211-006-0034-1","article-title":"A mixed finite volume scheme for anisotropic diffusion problems on any grid","volume":"105","author":"Droniou, J\u00e9r\u00f4me","year":"2006","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"33","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1201\/b16924-18","article-title":"An error estimator for nonconforming approximations of a nonlinear problem","author":"Dur\u00e1n, Ricardo","year":"1994"},{"issue":"6","key":"34","doi-asserted-by":"publisher","first-page":"903","DOI":"10.1051\/m2an:2004044","article-title":"Residual and hierarchical a posteriori error estimates for nonconforming mixed finite element methods","volume":"38","author":"El Alaoui, Linda","year":"2004","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"35","doi-asserted-by":"crossref","unstructured":"Alexandre Ern, Annette F. Stephansen, and Martin Vohral\u00edk, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection\u2013diffusion\u2013reaction problems, J. Comput. Appl. Math. 234 (2010), no. 1, 114\u2013130.","DOI":"10.1016\/j.cam.2009.12.009"},{"key":"36","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1086\/phos.67.4.188705","article-title":"Finite volume methods","author":"Eymard, Robert","year":"2000"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s00211-006-0036-z","article-title":"A combined finite volume\u2013nonconforming\/mixed-hybrid finite element scheme for degenerate parabolic problems","volume":"105","author":"Eymard, Robert","year":"2006","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"205","key":"38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2153393","article-title":"Remarks on mixed finite element methods for problems with rough coefficients","volume":"62","author":"Falk, Richard S.","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"39","doi-asserted-by":"publisher","first-page":"1658","DOI":"10.1137\/S0036142994276992","article-title":"Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems","volume":"34","author":"Hoppe, Ronald H. W.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"40","doi-asserted-by":"publisher","first-page":"2374","DOI":"10.1137\/S0036142902405217","article-title":"A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems","volume":"41","author":"Karakashian, Ohannes A.","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"257","key":"41","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1090\/S0025-5718-06-01903-X","article-title":"A posteriori error analysis for locally conservative mixed methods","volume":"76","author":"Kim, Kwang Y.","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"42","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1023\/A:1025518113877","article-title":"Residual a posteriori error estimates for the mixed finite element method","volume":"7","author":"Kirby, Robert","year":"2003","journal-title":"Comput. Geosci.","ISSN":"https:\/\/id.crossref.org\/issn\/1420-0597","issn-type":"print"},{"issue":"3","key":"43","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1007\/s00211-007-0121-y","article-title":"A posteriori error estimates for mixed finite element approximations of elliptic problems","volume":"108","author":"Larson, Mats G.","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"256","key":"44","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.1090\/S0025-5718-06-01872-2","article-title":"Energy norm a posteriori error estimates for mixed finite element methods","volume":"75","author":"Lovadina, Carlo","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"45","first-page":"219","article-title":"An abstract theory for mixed approximations of second order elliptic problems","volume":"8","author":"Marini, L. Donatella","year":"1989","journal-title":"Mat. Apl. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0101-8205","issn-type":"print"},{"issue":"3","key":"46","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1137\/0722029","article-title":"An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method","volume":"22","author":"Marini, Luisa Donatella","year":"1985","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"47","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF01396415","article-title":"Mixed finite elements in \ud835\udc45\u00b3","volume":"35","author":"N\u00e9d\u00e9lec, J.-C.","year":"1980","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"48","first-page":"38","article-title":"Isotropic and anisotropic a posteriori error estimation of the mixed finite element method for second order operators in divergence form","volume":"23","author":"Nicaise, Serge","year":"2006","journal-title":"Electron. Trans. Numer. Anal."},{"key":"49","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1007\/BF00252910","article-title":"An optimal Poincar\u00e9 inequality for convex domains","volume":"5","author":"Payne, L. E.","year":"1960","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"50","unstructured":"Gergina V. Pencheva, Martin Vohral\u00edk, 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."},{"key":"51","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1090\/qam\/25902","article-title":"Approximations in elasticity based on the concept of function space","volume":"5","author":"Prager, W.","year":"1947","journal-title":"Quart. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-569X","issn-type":"print"},{"key":"52","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-85268-1","volume-title":"Numerical approximation of partial differential equations","volume":"23","author":"Quarteroni, Alfio","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540571116"},{"key":"53","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"issue":"4","key":"54","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1163\/156939805775122271","article-title":"Functional-type a posteriori error estimates for mixed finite element methods","volume":"20","author":"Repin, S. I.","year":"2005","journal-title":"Russian J. Numer. Anal. Math. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0927-6467","issn-type":"print"},{"issue":"3","key":"55","doi-asserted-by":"publisher","first-page":"928","DOI":"10.1137\/050641533","article-title":"Two-sided a posteriori error estimates for mixed formulations of elliptic problems","volume":"45","author":"Repin, Sergey","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"56","first-page":"523","article-title":"Mixed and hybrid methods","author":"Roberts, J. E.","year":"1991"},{"issue":"1","key":"57","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1051\/m2an\/1991250101511","article-title":"Postprocessing schemes for some mixed finite elements","volume":"25","author":"Stenberg, Rolf","year":"1991","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"58","unstructured":"R. Verf\u00fcrth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Teubner-Wiley, Stuttgart, 1996."},{"issue":"7-8","key":"59","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1080\/01630560500444533","article-title":"On the discrete Poincar\u00e9-Friedrichs inequalities for nonconforming approximations of the Sobolev space \ud835\udc3b\u00b9","volume":"26","author":"Vohral\u00edk, Martin","year":"2005","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"issue":"2","key":"60","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1051\/m2an:2006013","article-title":"Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes","volume":"40","author":"Vohral\u00edk, Martin","year":"2006","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"61","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1051\/proc:071806","article-title":"A posteriori error estimates for finite volume and mixed finite element discretizations of convection-diffusion-reaction equations","author":"Vohral\u00edk, Martin","year":"2007"},{"issue":"4","key":"62","doi-asserted-by":"publisher","first-page":"1570","DOI":"10.1137\/060653184","article-title":"A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations","volume":"45","author":"Vohral\u00edk, Martin","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"63","unstructured":"\\bysame, 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"},{"issue":"1","key":"64","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s00211-008-0168-4","article-title":"Residual flux-based a posteriori error estimates for finite volume and related locally conservative methods","volume":"111","author":"Vohral\u00edk, Martin","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"65","doi-asserted-by":"crossref","unstructured":"Martin Vohral\u00edk, Ji\u0159\u00ed Mary\u0161ka, and Otto Sever\u00fdn, Mixed and nonconforming finite element methods on a system of polygons, Appl. Numer. Math. 57 (2007), 176\u2013193.","DOI":"10.1016\/j.apnum.2006.02.005"},{"issue":"3","key":"66","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1137\/S0036142903431687","article-title":"A posteriori error estimates for the mortar mixed finite element method","volume":"43","author":"Wheeler, Mary F.","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"228","key":"67","doi-asserted-by":"publisher","first-page":"1347","DOI":"10.1090\/S0025-5718-99-01125-4","article-title":"A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements","volume":"68","author":"Wohlmuth, Barbara I.","year":"1999","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"68","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1002\/nme.874","article-title":"From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions","volume":"59","author":"Younes, Anis","year":"2004","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"2","key":"69","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1002\/nme.1620240206","article-title":"A simple error estimator and adaptive procedure for practical engineering analysis","volume":"24","author":"Zienkiewicz, O. C.","year":"1987","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-2010-02375-0\/S0025-5718-2010-02375-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-2010-02375-0\/S0025-5718-2010-02375-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T04:03:25Z","timestamp":1627617805000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-272\/S0025-5718-2010-02375-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,26]]},"references-count":69,"journal-issue":{"issue":"272","published-print":{"date-parts":[[2010,10]]}},"alternative-id":["S0025-5718-2010-02375-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2010-02375-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,5,26]]}}}