{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T08:44:14Z","timestamp":1772700254764,"version":"3.50.1"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100003406","name":"Tekes","doi-asserted-by":"publisher","award":["3305\/31\/2015"],"award-info":[{"award-number":["3305\/31\/2015"]}],"id":[{"id":"10.13039\/501100003406","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We discuss the differences between the penalty, mixed and stabilised methods for the finite element approximation of the obstacle problem. The theoretical properties of the methods are discussed and illustrated through numerical examples.<\/jats:p>","DOI":"10.1515\/cmam-2017-0011","type":"journal-article","created":{"date-parts":[[2017,6,8]],"date-time":"2017-06-08T10:02:04Z","timestamp":1496916124000},"page":"413-429","source":"Crossref","is-referenced-by-count":8,"title":["On Finite Element Formulations for the Obstacle Problem \u2013 Mixed and Stabilised Methods"],"prefix":"10.1515","volume":"17","author":[{"given":"Tom","family":"Gustafsson","sequence":"first","affiliation":[{"name":"Department of Mathematics and Systems Analysis , Aalto University School of Science , Espoo , Finland"}]},{"given":"Rolf","family":"Stenberg","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Systems Analysis , Aalto University School of Science , Espoo , Finland"}]},{"given":"Juha","family":"Videman","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Instituto Superior T\u00e9cnico, Universidade de Lisboa , Lisboa , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2017,6,8]]},"reference":[{"key":"2023033114491590918_j_cmam-2017-0011_ref_001_w2aab3b7e1593b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka,\nError-bounds for finite element method,\nNumer. Math. 16 (1970\/71), 322\u2013333.","DOI":"10.1007\/BF02165003"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_002_w2aab3b7e1593b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka,\nThe finite element method with Lagrangian multipliers,\nNumer. Math. 20 (1973), 179\u2013192.","DOI":"10.1007\/BF01436561"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_003_w2aab3b7e1593b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"H. J. C. Barbosa and T. J. R. Hughes,\nThe finite element method with Lagrange multipliers on the boundary: Circumventing the Babu\u0161ka\u2013Brezzi condition,\nComput. Methods Appl. Mech. Engrg. 85 (1991), 109\u2013128.","DOI":"10.1016\/0045-7825(91)90125-P"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_004_w2aab3b7e1593b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"R. Becker, P. Hansbo and R. Stenberg,\nA finite element method for domain decomposition with non-matching grids,\nM2AN Math. Model. Numer. Anal. 37 (2003), 209\u2013225.","DOI":"10.1051\/m2an:2003023"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_005_w2aab3b7e1593b1b6b1ab2ab5Aa","unstructured":"D. Braess,\nFinite Elements, 3rd ed.,\nCambridge University Press, Cambridge, 2007."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_006_w2aab3b7e1593b1b6b1ab2ab6Aa","unstructured":"D. Braess, C. Carstensen and R. H. W. Hoppe,\nError reduction in adaptive finite element approximations of elliptic obstacle problems,\nJ. Comput. Math. 27 (2009), 148\u2013169."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_007_w2aab3b7e1593b1b6b1ab2ab7Aa","unstructured":"H. Brezis,\nNouveaux th\u00e9or\u00e8mes de r\u00e9gularit\u00e9 pour les probl\u00e8mes unilat\u00e9raux,\nLes Rencontres Physiciens-Math\u00e9maticiens de Strasbourg \u2013 RCP 25,\nUniversit\u00e9 Louis Pasteur, Strasbourg (1971), 1\u201314."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_008_w2aab3b7e1593b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"F. Brezzi,\nOn the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,\nRev. Fran\u00e7aise Automat. Informat. Rech. Op\u00e9r. S\u00e9r. Rouge 8 (1974), 129\u2013151.","DOI":"10.1051\/m2an\/197408R201291"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_009_w2aab3b7e1593b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"F. Brezzi, W. W. Hager and P.-A. Raviart,\nError estimates for the finite element solution of variational inequalities. II: Mixed methods,\nNumer. Math. 31 (1978\/79), 1\u201316.","DOI":"10.1007\/BF01396010"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_010_w2aab3b7e1593b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"F. Brezzi and J. Pitk\u00e4ranta,\nOn the stabilization of finite element approximations of the Stokes equations,\nEfficient Solutions of Elliptic Systems (Kiel 1984),\nNotes Numer. Fluid Mech. 10,\nFriedr. Vieweg, Braunschweig (1984), 11\u201319.","DOI":"10.1007\/978-3-663-14169-3_2"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_011_w2aab3b7e1593b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"E. Burman, P. Hansbo, M. G. Larson and R. Stenberg,\nGalerkin least squares finite element method for the obstacle problem,\nComput. Methods Appl. Mech. Engrg. 313 (2016), 362\u2013374.","DOI":"10.1016\/j.cma.2016.09.025"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_012_w2aab3b7e1593b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"F. Chouly and P. Hild,\nA Nitsche-based method for unilateral contact problems: Numerical analysis,\nSIAM J. Numer. Anal. 51 (2013), 1295\u20131307.","DOI":"10.1137\/12088344X"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_013_w2aab3b7e1593b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStudies Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_014_w2aab3b7e1593b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"R. S. Falk,\nError estimates for the approximation of a class of variational inequalities,\nMath. Comp. 28 (1974), 963\u2013971.","DOI":"10.1090\/S0025-5718-1974-0391502-8"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_015_w2aab3b7e1593b1b6b1ab2ac15Aa","unstructured":"R. Glowinski,\nNumerical Methods for Nonlinear Variational Problems,\nSci. Comput.,\nSpringer, Berlin, 2008."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_016_w2aab3b7e1593b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"R. Glowinski and P. Le Tallec,\nAugmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,\nSIAM Stud. Appl. Math. 9,\nSociety for Industrial and Applied Mathematics, Philadelphia, 1989.","DOI":"10.1137\/1.9781611970838"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_017_w2aab3b7e1593b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"T. Gudi,\nA new error analysis for discontinuous finite element methods for linear elliptic problems,\nMath. Comp. 79 (2010), 2169\u20132189.","DOI":"10.1090\/S0025-5718-10-02360-4"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_018_w2aab3b7e1593b1b6b1ab2ac18Aa","unstructured":"T. Gustafsson, R. Stenberg and J. Videman,\nMixed and stabilized finite element methods for the obstacle problem,\nSIAM J. Numer. Anal., to appear."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_019_w2aab3b7e1593b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"P. Hild and Y. Renard,\nA stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics,\nNumer. Math. 115 (2010), 101\u2013129.","DOI":"10.1007\/s00211-009-0273-z"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_020_w2aab3b7e1593b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"M. Hinterm\u00fcller, K. Ito and K. Kunisch,\nThe primal-dual active set strategy as semismooth newton method,\nSIAM J. Optim. 13 (2003), 865\u2013888.","DOI":"10.1137\/S1052623401383558"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_021_w2aab3b7e1593b1b6b1ab2ac21Aa","doi-asserted-by":"crossref","unstructured":"I. Hlav\u00e1\u010dek, J. Haslinger, J. Ne\u010das and J. Lov\u00ed\u0161ek,\nSolution of Variational Inequalities in Mechanics,\nAppl. Math. Sci. 66,\nSpringer, New York, 1988.","DOI":"10.1007\/978-1-4612-1048-1"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_022_w2aab3b7e1593b1b6b1ab2ac22Aa","doi-asserted-by":"crossref","unstructured":"T. J. R. Hughes, L. P. Franca and M. Balestra,\nA new finite element formulation for computational fluid dynamics. V. Circumventing the Babu\u0161ka\u2013Brezzi condition: A stable Petrov\u2013Galerkin formulation of the Stokes problem accommodating equal-order interpolations,\nComput. Methods Appl. Mech. Engrg. 59 (1986), 85\u201399.","DOI":"10.1016\/0045-7825(86)90025-3"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_023_w2aab3b7e1593b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"C. Johnson,\nAdaptive finite element methods for the obstacle problem,\nMath. Models Methods Appl. Sci. 2 (1992), 483\u2013487.","DOI":"10.1142\/S0218202592000284"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_024_w2aab3b7e1593b1b6b1ab2ac24Aa","doi-asserted-by":"crossref","unstructured":"M. Juntunen and R. Stenberg,\nNitsche\u2019s method for general boundary conditions,\nMath. Comp. 78 (2009), 1353\u20131374.","DOI":"10.1090\/S0025-5718-08-02183-2"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_025_w2aab3b7e1593b1b6b1ab2ac25Aa","unstructured":"J.-L. Lions,\nQuelques M\u00e9thodes de R\u00e9solution des Probl\u00e8mes aux Limites Non Lin\u00e9aires,\nDunod, Paris, 1969."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_026_w2aab3b7e1593b1b6b1ab2ac26Aa","doi-asserted-by":"crossref","unstructured":"J.-L. Lions and G. Stampacchia,\nVariational inequalities,\nComm. Pure Appl. Math. 20 (1967), 493\u2013519.","DOI":"10.1002\/cpa.3160200302"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_027_w2aab3b7e1593b1b6b1ab2ac27Aa","unstructured":"N. L\u00fcthen, M. Juntunen and R. Stenberg,\nAn improved a priori error analysis of Nitsche\u2019s method for Robin boundary conditions,\npreprint (2015), https:\/\/arxiv.org\/abs\/1502.06515."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_028_w2aab3b7e1593b1b6b1ab2ac28Aa","doi-asserted-by":"crossref","unstructured":"J. Nitsche,\n\u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet-Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind,\nAbh. Math. Semin. Univ. Hambg. 36 (1971), 9\u201315.","DOI":"10.1007\/BF02995904"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_029_w2aab3b7e1593b1b6b1ab2ac29Aa","doi-asserted-by":"crossref","unstructured":"R. Nochetto, K. Siebert and A. Veeser,\nPointwise a posteriori error control for elliptic obstacle problems,\nNumer. Math. 95 (2003), 163\u2013195.","DOI":"10.1007\/s00211-002-0411-3"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_030_w2aab3b7e1593b1b6b1ab2ac30Aa","doi-asserted-by":"crossref","unstructured":"R. Pierre,\nSimple C0{C^{0}} approximations for the computation of incompressible flows,\nComput. Methods Appl. Mech. Engrg. 68 (1988), 205\u2013227.","DOI":"10.1016\/0045-7825(88)90116-8"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_031_w2aab3b7e1593b1b6b1ab2ac31Aa","doi-asserted-by":"crossref","unstructured":"J. Pitk\u00e4ranta,\nAnalysis of some low-order finite element schemes for Mindlin\u2013Reissner and Kirchhoff plates,\nNumer. Math. 53 (1988), 237\u2013254.","DOI":"10.1007\/BF01395887"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_032_w2aab3b7e1593b1b6b1ab2ac32Aa","doi-asserted-by":"crossref","unstructured":"R. Scholz,\nNumerical solution of the obstacle problem by the penalty method,\nComputing 32 (1984), 297\u2013306.","DOI":"10.1007\/BF02243774"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_033_w2aab3b7e1593b1b6b1ab2ac33Aa","unstructured":"R. Stenberg,\nMortaring by a method of J. A. Nitsche,\nComputational Mechanics. New Trends and Applications,\nCIMNE, Barcelona (1988), 1\u20136."},{"key":"2023033114491590918_j_cmam-2017-0011_ref_034_w2aab3b7e1593b1b6b1ab2ac34Aa","doi-asserted-by":"crossref","unstructured":"R. Stenberg,\nOn some techniques for approximating boundary conditions in the finite element method,\nJ. Comput. Appl. Math. 63 (1995), 139\u2013148.","DOI":"10.1016\/0377-0427(95)00057-7"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_035_w2aab3b7e1593b1b6b1ab2ac35Aa","doi-asserted-by":"crossref","unstructured":"M. Ulbrich,\nSemismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces,\nMOS-SIAM Ser. Optim.,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2011.","DOI":"10.1137\/1.9781611970692"},{"key":"2023033114491590918_j_cmam-2017-0011_ref_036_w2aab3b7e1593b1b6b1ab2ac36Aa","doi-asserted-by":"crossref","unstructured":"Q. Zou, A. Veeser, R. Kornhuber and C. Gr\u00e4ser,\nHierarchical error estimates for the energy functional in obstacle problems,\nNumer. Math. 117 (2011), 653\u2013677.","DOI":"10.1007\/s00211-011-0364-5"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/3\/article-p413.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0011\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0011\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:18:25Z","timestamp":1680297505000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0011\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,8]]},"references-count":36,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,6,17]]},"published-print":{"date-parts":[[2017,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0011"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0011","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,6,8]]}}}