{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T10:40:08Z","timestamp":1778582408679,"version":"3.51.4"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem.\nThe method comprises of the Crouzeix\u2013Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization.\nWe derive an error estimate of optimal order <jats:inline-formula id=\"j_cmam-2019-0057_ineq_9999_w2aab3b7e3887b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\ud835\udcaa<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>h<\/m:mi>\n                                    <m:mo>+<\/m:mo>\n                                    <m:mrow>\n                                       <m:mi>\u0394<\/m:mi>\n                                       <m:mo>\u2062<\/m:mo>\n                                       <m:mi>t<\/m:mi>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{\\mathcal{O}(h+\\Delta t)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nin a certain energy norm defined precisely in the article. We only assume the realistic regularity <jats:inline-formula id=\"j_cmam-2019-0057_ineq_9998_w2aab3b7e3887b1b6b1aab1c14b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi>u<\/m:mi>\n                                 <m:mi>t<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:msup>\n                                    <m:mi>L<\/m:mi>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mn>0<\/m:mn>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mi>T<\/m:mi>\n                                    <m:mo>;<\/m:mo>\n                                    <m:mrow>\n                                       <m:msup>\n                                          <m:mi>L<\/m:mi>\n                                          <m:mn>2<\/m:mn>\n                                       <\/m:msup>\n                                       <m:mo>\u2062<\/m:mo>\n                                       <m:mrow>\n                                          <m:mo>(<\/m:mo>\n                                          <m:mi>\u03a9<\/m:mi>\n                                          <m:mo>)<\/m:mo>\n                                       <\/m:mrow>\n                                    <\/m:mrow>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{u_{t}\\in L^{2}(0,T;L^{2}(\\Omega))}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.<\/jats:p>","DOI":"10.1515\/cmam-2019-0057","type":"journal-article","created":{"date-parts":[[2019,8,28]],"date-time":"2019-08-28T09:59:41Z","timestamp":1566986381000},"page":"273-292","source":"Crossref","is-referenced-by-count":3,"title":["Crouzeix\u2013Raviart Finite Element Approximation for the Parabolic Obstacle Problem"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2093-1382","authenticated-orcid":false,"given":"Thirupathi","family":"Gudi","sequence":"first","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}]},{"given":"Papri","family":"Majumder","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Science , Bangalore 560012 , India"}]}],"member":"374","published-online":{"date-parts":[[2019,8,28]]},"reference":[{"key":"2023033110443917614_j_cmam-2019-0057_ref_001_w2aab3b7e3887b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"J.  Alberty, C.  Carstensen and S. A.  Funken,\nRemarks around 50 lines of Matlab: Short finite element implementation,\nNumer. Algorithms 20 (1999), no. 2\u20133, 117\u2013137.","DOI":"10.1023\/A:1019155918070"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_002_w2aab3b7e3887b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"C.  Bahriawati and C.  Carstensen,\nThree MATLAB implementations of the lowest-order Raviart\u2013Thomas MFEM with a posteriori error control,\nComput. Methods Appl. Math. 5 (2005), no. 4, 333\u2013361.","DOI":"10.2478\/cmam-2005-0016"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_003_w2aab3b7e3887b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"L.  Banz and E. P.  Stephan,\nhp-adaptive IPDG\/TDG-FEM for parabolic obstacle problems,\nComput. Math. Appl. 67 (2014), no. 4, 712\u2013731.","DOI":"10.1016\/j.camwa.2013.03.003"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_004_w2aab3b7e3887b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"A. E.  Berger and R. S.  Falk,\nAn error estimate for the truncation method for the solution of parabolic obstacle variational inequalities,\nMath. Comp. 31 (1977), no. 139, 619\u2013628.","DOI":"10.1090\/S0025-5718-1977-0438707-8"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_005_w2aab3b7e3887b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_006_w2aab3b7e3887b1b6b1ab2ab6Aa","unstructured":"H.  Br\u00e9zis,\nProbl\u00e8mes unilat\u00e9raux,\nJ. Math. Pures Appl. (9) 51 (1972), 1\u2013168."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_007_w2aab3b7e3887b1b6b1ab2ab7Aa","unstructured":"H.  Br\u00e9zis,\nOp\u00e9rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert,\nNorth-Holland Math. Stud. 5,\nNorth-Holland, Amsterdam, 1973."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_008_w2aab3b7e3887b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, J.  Gedicke and D.  Rim,\nExplicit error estimates for Courant, Crouzeix\u2013Raviart and Raviart\u2013Thomas finite element methods,\nJ. Comput. Math. 30 (2012), no. 4, 337\u2013353.","DOI":"10.4208\/jcm.1108-m3677"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_009_w2aab3b7e3887b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and K.  K\u00f6hler,\nNonconforming FEM for the obstacle problem,\nIMA J. Numer. Anal. 37 (2017), no. 1, 64\u201393.","DOI":"10.1093\/imanum\/drw005"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_010_w2aab3b7e3887b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"Z.  Chen and R. H.  Nochetto,\nResidual type a posteriori error estimates for elliptic obstacle problems,\nNumer. Math. 84 (2000), no. 4, 527\u2013548.","DOI":"10.1007\/s002110050009"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_011_w2aab3b7e3887b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_012_w2aab3b7e3887b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"M.  Crouzeix and P.-A.  Raviart,\nConforming and nonconforming finite element methods for solving the stationary Stokes equations. I,\nRev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge 7 (1973), no. R-3, 33\u201375.","DOI":"10.1051\/m2an\/197307R300331"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_013_w2aab3b7e3887b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"S.  Gaddam and T.  Gudi,\nInhomogeneous Dirichlet boundary condition in the a posteriori error control of the obstacle problem,\nComput. Math. Appl. 75 (2018), no. 7, 2311\u20132327.","DOI":"10.1016\/j.camwa.2017.12.010"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_014_w2aab3b7e3887b1b6b1ab2ac14Aa","unstructured":"R.  Glowinski, J. L.  Lions and R.  Th\u00e9moli\u00e9res,\nNumerical Methods for Variational Inequalities,\nNorth-Holland, Amsterdam, 1981."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_015_w2aab3b7e3887b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"T.  Gudi and P.  Majumder,\nConforming and discontinuous Galerkin FEM in space for solving parabolic obstacle problem,\nComput. Math. Appl. 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Math. 52,\nHindustan Book, New Delhi, 2009.","DOI":"10.1007\/978-93-86279-42-2"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_022_w2aab3b7e3887b1b6b1ab2ac22Aa","doi-asserted-by":"crossref","unstructured":"D.  Kinderlehrer and G.  Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications,\nClass. Appl. Math. 31,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000.","DOI":"10.1137\/1.9780898719451"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_023_w2aab3b7e3887b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"J. L.  Lions,\nPartial differential inequalities,\nUspehi Mat. Nauk 26 (1971), no. 2(158), 205\u2013263;\ntranslation in Russian Math. Surveys 27, no. 2, 91\u2013159.","DOI":"10.1070\/RM1972v027n02ABEH001373"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_024_w2aab3b7e3887b1b6b1ab2ac24Aa","doi-asserted-by":"crossref","unstructured":"J.-L.  Lions and G.  Stampacchia,\nVariational inequalities,\nComm. Pure Appl. 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Verdi,\nA posteriori error estimates for variable time-step discretizations of nonlinear evolution equations,\nComm. Pure Appl. Math. 53 (2000), no. 5, 525\u2013589.","DOI":"10.1002\/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_028_w2aab3b7e3887b1b6b1ab2ac28Aa","doi-asserted-by":"crossref","unstructured":"E.  Ot\u00e1rola and A. J.  Salgado,\nFinite element approximation of the parabolic fractional obstacle problem,\nSIAM J. Numer. Anal. 54 (2016), no. 4, 2619\u20132639.","DOI":"10.1137\/15M1029801"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_029_w2aab3b7e3887b1b6b1ab2ac29Aa","doi-asserted-by":"crossref","unstructured":"A. K.  Pani and P. C.  Das,\nA priori error estimates for a single-phase quasilinear Stefan problem in one space dimension,\nIMA J. Numer. Anal. 11 (1991), no. 3, 377\u2013392.","DOI":"10.1093\/imanum\/11.3.377"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_030_w2aab3b7e3887b1b6b1ab2ac30Aa","doi-asserted-by":"crossref","unstructured":"J.  Rulla,\nError analysis for implicit approximations to solutions to Cauchy problems,\nSIAM J. Numer. Anal. 33 (1996), no. 1, 68\u201387.","DOI":"10.1137\/0733005"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_031_w2aab3b7e3887b1b6b1ab2ac31Aa","unstructured":"G.  Savar\u00e9,\nWeak solutions and maximal regularity for abstract evolution inequalities,\nAdv. Math. Sci. Appl. 6 (1996), no. 2, 377\u2013418."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_032_w2aab3b7e3887b1b6b1ab2ac32Aa","unstructured":"G.  Stampacchia,\n\u00c8quations elliptiques du second ordre \u00e0 coefficients discontinus,\nS\u00e9min. Math. Sup\u00e9r. 16,\nLes Presses de l\u2019Universit\u00e9 de Montr\u00e9al, Montreal, 1966."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_033_w2aab3b7e3887b1b6b1ab2ac33Aa","unstructured":"V.  Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer Ser. Comput. Math. 25,\nSpringer, Berlin, 2006."},{"key":"2023033110443917614_j_cmam-2019-0057_ref_034_w2aab3b7e3887b1b6b1ab2ac34Aa","doi-asserted-by":"crossref","unstructured":"C.  Vuik,\nAn L2L^{2}-error estimate for an approximation of the solution of a parabolic variational inequality,\nNumer. Math. 57 (1990), no. 5, 453\u2013471.","DOI":"10.1007\/BF01386423"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_035_w2aab3b7e3887b1b6b1ab2ac35Aa","doi-asserted-by":"crossref","unstructured":"X.  Yang, G.  Wang and X.  Gu,\nNumerical solution for a parabolic obstacle problem with nonsmooth initial data,\nNumer. Methods Partial Differential Equations 30 (2014), no. 5, 1740\u20131754.","DOI":"10.1002\/num.21893"},{"key":"2023033110443917614_j_cmam-2019-0057_ref_036_w2aab3b7e3887b1b6b1ab2ac36Aa","unstructured":"C.-S.  Zhang,\nAdaptive Finite Element Methods for Variational Inequalities: Theory and Applications in Finance,\nProQuest LLC, Ann Arbor, 2007;\nThesis (Ph.D.)\u2013University of Maryland, College Park."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/2\/article-p273.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0057\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0057\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:51:56Z","timestamp":1680267116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2019-0057\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,28]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,8,14]]},"published-print":{"date-parts":[[2020,4,1]]}},"alternative-id":["10.1515\/cmam-2019-0057"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2019-0057","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2019,8,28]]}}}