{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:18:14Z","timestamp":1773249494832,"version":"3.50.1"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement\napproximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable\ncombination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction\nprocedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin\n[2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect\nto the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in\nnext-to-lowest order Raviart\u2013Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each\nelement; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the\none for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the\nincompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations\ninvolving different Lam\u00e9 parameters including the incompressible limit case.<\/jats:p>","DOI":"10.1515\/cmam-2018-0004","type":"journal-article","created":{"date-parts":[[2018,4,12]],"date-time":"2018-04-12T08:05:35Z","timestamp":1523520335000},"page":"663-679","source":"Crossref","is-referenced-by-count":11,"title":["A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction"],"prefix":"10.1515","volume":"19","author":[{"given":"Fleurianne","family":"Bertrand","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Duisburg\u2013Essen , Thea-Leymann-Str. 9, 45127 Essen , Germany"}]},{"given":"Marcel","family":"Moldenhauer","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Duisburg\u2013Essen , Thea-Leymann-Str. 9, 45127 Essen , Germany"}]},{"given":"Gerhard","family":"Starke","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Duisburg\u2013Essen , Thea-Leymann-Str. 9, 45127 Essen , Germany"}]}],"member":"374","published-online":{"date-parts":[[2018,3,7]]},"reference":[{"key":"2023033110340711501_j_cmam-2018-0004_ref_001_w2aab3b7d410b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"B.  Achdou, F.  Bernardi and F.  Coquel,\nA priori and a posteriori analysis of finite volume discretizations of Darcy\u2019s equations,\nNumer. Math. 96 (2003), 17\u201342.","DOI":"10.1007\/s00211-002-0436-7"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_002_w2aab3b7d410b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth, A.  Allendes, G. R.  Barrenechea and R.  Rankin,\nComputable error bounds for nonconforming Fortin\u2013Soulie finite element approximation of the Stokes problem,\nIMA J. Numer. Anal. 32 (2012), 417\u2013447.","DOI":"10.1093\/imanum\/drr006"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_003_w2aab3b7d410b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth and R.  Rankin,\nRobust a posteriori error estimation for the nonconforming Fortin\u2013Soulie finite element approximation,\nMath. Comp. 77 (2008), 1917\u20131939.","DOI":"10.1090\/S0025-5718-08-02116-9"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_004_w2aab3b7d410b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"M.  Ainsworth and R.  Rankin,\nGuaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity,\nInt. J. Numer. Methods Eng. 82 (2010), 1114\u20131157.","DOI":"10.1002\/nme.2799"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_005_w2aab3b7d410b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold, G.  Awanou and R.  Winther,\nFinite elements for symmetric tensors in three dimensions,\nMath. Comp. 77 (2008), 1229\u20131251.","DOI":"10.1090\/S0025-5718-08-02071-1"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_006_w2aab3b7d410b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold and R.  Winther,\nMixed finite elements for elasticity,\nNumer. Math. 92 (2002), 401\u2013419.","DOI":"10.1007\/s002110100348"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_007_w2aab3b7d410b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"D.  Boffi, F.  Brezzi and M.  Fortin,\nMixed Finite Element Methods and Applications,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_008_w2aab3b7d410b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner,\nKorn\u2019s inequalities for piecewise H1{H^{1}} vector fields,\nMath. Comp. 73 (2003), 1067\u20131087.","DOI":"10.1090\/S0025-5718-03-01579-5"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_009_w2aab3b7d410b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"Z.  Cai and G.  Starke,\nLeast squares methods for linear elasticity,\nSIAM J. Numer. Anal. 42 (2004), 826\u2013842.","DOI":"10.1137\/S0036142902418357"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_010_w2aab3b7d410b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and G.  Dolzmann,\nA posteriori error estimates for mixed FEM in elasticity,\nNumer. Math. 81 (1998), 187\u2013209.","DOI":"10.1007\/s002110050389"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_011_w2aab3b7d410b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and M.  Vohral\u00edk,\nPolynomial-degree-robust a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations,\nSIAM J. Numer. Anal. 53 (2015), 1058\u20131081.","DOI":"10.1137\/130950100"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_012_w2aab3b7d410b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"M.  Fortin,\nA three-dimensional quadratic nonconforming element,\nNumer. Math. 46 (1985), 269\u2013279.","DOI":"10.1007\/BF01390424"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_013_w2aab3b7d410b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"M.  Fortin and M.  Soulie,\nA non-conforming piecewise quadratic finite element on triangles,\nInt. J. Numer. Methods Eng. 19 (1983), 505\u2013520.","DOI":"10.1002\/nme.1620190405"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_014_w2aab3b7d410b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"V.  Girault and P.  Raviart,\nFinite Element Methods for Navier\u2013Stokes Equations,\nSpringer, New York, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_015_w2aab3b7d410b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"A.  Hannukainen, R.  Stenberg and M.  Vohral\u00edk,\nA unified framework for a posteriori error estimation for the Stokes equation,\nNumer. Math. 122 (2012), 725\u2013769.","DOI":"10.1007\/s00211-012-0472-x"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_016_w2aab3b7d410b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"C. O.  Horgan,\nKorn\u2019s inequalities and their applications in continuum mechanics,\nSIAM Rev. 37 (1995), 491\u2013511.","DOI":"10.1137\/1037123"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_017_w2aab3b7d410b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"K.-Y.  Kim,\nGuaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry,\nSIAM J. Numer. Anal. 49 (2011), 2364\u20132385.","DOI":"10.1137\/110823031"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_018_w2aab3b7d410b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"K.-Y.  Kim,\nFlux reconstruction for the P2 nonconforming finite element method with application to a posteriori error estimation,\nAppl. Numer. Math. 62 (2012), 1701\u20131717.","DOI":"10.1016\/j.apnum.2012.06.027"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_019_w2aab3b7d410b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"P.  Ladev\u00e8ze and D.  Leguillon,\nError estimate procedure in the finite element method and applications,\nSIAM J. Numer. Anal. 20 (1983), 485\u2013509.","DOI":"10.1137\/0720033"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_020_w2aab3b7d410b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"B.  M\u00fcller and G.  Starke,\nStress-based finite element methods in linear and nonlinear solid mechanics,\nAdvanced Finite Element Technologies,\nCISM Courses and Lectures 566,\nSpringer, Cham (2016), 69\u2013104.","DOI":"10.1007\/978-3-319-31925-4_4"},{"key":"2023033110340711501_j_cmam-2018-0004_ref_021_w2aab3b7d410b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"S.  Nicaise, K.  Witowski and B.  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Verf\u00fcrth,\nA Posteriori Error Estimation Techniques for Finite Element Methods,\nOxford University Press, New York, 2013.","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/19\/3\/article-p663.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0004\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0004\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:00:37Z","timestamp":1680264037000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0004\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,7]]},"references-count":23,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,6,27]]},"published-print":{"date-parts":[[2019,7,1]]}},"alternative-id":["10.1515\/cmam-2018-0004"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0004","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2018,3,7]]}}}