{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T15:12:58Z","timestamp":1760800378085},"reference-count":35,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/501100001843","name":"Science and Engineering Research Board","doi-asserted-by":"publisher","award":["MTR\/2017\/000199"],"award-info":[{"award-number":["MTR\/2017\/000199"]}],"id":[{"id":"10.13039\/501100001843","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Consider the distributed optimal control problem governed by the von K\u00e1rm\u00e1n equations defined on a polygonal domain of <jats:inline-formula id=\"j_cmam-2020-0030_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>\u211d<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2020-0030_eq_0764.png\" \/>\n                        <jats:tex-math>{\\mathbb{R}^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> that describe the deflection of very thin plates with box constraints on the control variable.\nThis article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables.\nThe control is discretized using piecewise constants.\nA priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution.\nError estimates in lower-order norms for the state and adjoint variables are derived.\nThe lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable.\nNumerical results confirm the theoretical results obtained.<\/jats:p>","DOI":"10.1515\/cmam-2020-0030","type":"journal-article","created":{"date-parts":[[2020,5,26]],"date-time":"2020-05-26T16:33:00Z","timestamp":1590510780000},"page":"233-262","source":"Crossref","is-referenced-by-count":5,"title":["Morley FEM for a Distributed Optimal Control Problem Governed by the von K\u00e1rm\u00e1n Equations"],"prefix":"10.1515","volume":"21","author":[{"given":"Sudipto","family":"Chowdhury","sequence":"first","affiliation":[{"name":"Department of Mathematics , Indian Institute of Technology Bombay , Powai , Mumbai 400076 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Neela","family":"Nataraj","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Technology Bombay , Powai , Mumbai 400076 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Devika","family":"Shylaja","sequence":"additional","affiliation":[{"name":"IITB-Monash Research Academy , Indian Institute of Technology Bombay , Powai , Mumbai 400076 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,4,15]]},"reference":[{"key":"2023033111222112637_j_cmam-2020-0030_ref_001","doi-asserted-by":"crossref","unstructured":"M. S.  Berger,\nOn von K\u00e1rm\u00e1n\u2019s equations and the buckling of a thin elastic plate. I. The clamped plate,\nComm. Pure Appl. Math. 20 (1967), 687\u2013719.","DOI":"10.1002\/cpa.3160200405"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_002","doi-asserted-by":"crossref","unstructured":"M. S.  Berger and P. C.  Fife,\nOn von Karman\u2019s equations and the buckling of a thin elastic plate,\nBull. Amer. Math. Soc. 72 (1966), 1006\u20131011.","DOI":"10.1090\/S0002-9904-1966-11620-8"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_003","doi-asserted-by":"crossref","unstructured":"M. S.  Berger and P. C.  Fife,\nVon K\u00e1rm\u00e1n\u2019s equations and the buckling of a thin elastic plate. II. Plate with general edge conditions,\nComm. Pure Appl. Math. 21 (1968), 227\u2013241.","DOI":"10.1002\/cpa.3160210303"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_004","doi-asserted-by":"crossref","unstructured":"H.  Blum and R.  Rannacher,\nOn the boundary value problem of the biharmonic operator on domains with angular corners,\nMath. Methods Appl. Sci. 2 (1980), no. 4, 556\u2013581.","DOI":"10.1002\/mma.1670020416"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_005","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, M.  Neilan, A.  Reiser and L.-Y.  Sung,\nA \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     {C^{0}}\n                  \n                interior penalty method for a von K\u00e1rm\u00e1n plate,\nNumer. Math. 135 (2017), no. 3, 803\u2013832.","DOI":"10.1007\/s00211-016-0817-y"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_006","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L.-Y.  Sung,\n\n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     {C^{0}}\n                  \n                interior penalty methods for fourth order elliptic boundary value problems on polygonal domains,\nJ. Sci. Comput. 22\/23 (2005), 83\u2013118.","DOI":"10.1007\/s10915-004-4135-7"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_007","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, L.-Y.  Sung, H.  Zhang and Y.  Zhang,\nA Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates,\nJ. Comput. Appl. Math. 254 (2013), 31\u201342.","DOI":"10.1016\/j.cam.2013.02.028"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_008","doi-asserted-by":"crossref","unstructured":"F.  Brezzi,\nFinite element approximations of the von K\u00e1rm\u00e1n equations,\nRAIRO Anal. Num\u00e9r. 12 (1978), no. 4, 303\u2013312.","DOI":"10.1051\/m2an\/1978120403031"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_009","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and D.  Gallistl,\nGuaranteed lower eigenvalue bounds for the biharmonic equation,\nNumer. Math. 126 (2014), no. 1, 33\u201351.","DOI":"10.1007\/s00211-013-0559-z"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_010","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, D.  Gallistl and J.  Hu,\nA discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes,\nComput. Math. Appl. 68 (2014), no. 12, 2167\u20132181.","DOI":"10.1016\/j.camwa.2014.07.019"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_011","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, G.  Mallik and N.  Nataraj,\nA priori and a posteriori error control of discontinuous Galerkin finite element methods for the von K\u00e1rm\u00e1n equations,\nIMA J. Numer. Anal. 39 (2019), no. 1, 167\u2013200.","DOI":"10.1093\/imanum\/dry003"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_012","doi-asserted-by":"crossref","unstructured":"C.  Carstensen, G.  Mallik and N.  Nataraj,\nNonconforming finite element discretization for semilinear problems with trilinear nonlinearity,\nIMA J. Numer. Anal. (2020), to appear.","DOI":"10.1093\/imanum\/drz071"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_013","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and S.  Puttkammer,\nHow to prove the discrete reliability for nonconforming finite element methods,\nJ. Comput. Math. 38 (2020), 142\u2013175.","DOI":"10.4208\/jcm.1908-m2018-0174"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_014","doi-asserted-by":"crossref","unstructured":"E.  Casas, M.  Mateos and J.-P.  Raymond,\nError estimates for the numerical approximation of a distributed control problem for the steady-state Navier\u2013Stokes equations,\nSIAM J. Control Optim. 46 (2007), no. 3, 952\u2013982.","DOI":"10.1137\/060649999"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_015","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":"2023033111222112637_j_cmam-2020-0030_ref_016","unstructured":"P. G.  Ciarlet,\nMathematical Elasticity. Vol. II. Theory of Plates,\nStud. Math. Appl. 27,\nNorth-Holland, Amsterdam, 1997."},{"key":"2023033111222112637_j_cmam-2020-0030_ref_017","doi-asserted-by":"crossref","unstructured":"D.  Gallistl,\nMorley finite element method for the eigenvalues of the biharmonic operator,\nIMA J. Numer. Anal. 35 (2015), no. 4, 1779\u20131811.","DOI":"10.1093\/imanum\/dru054"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_018","unstructured":"P.  Grisvard,\nSingularities in Boundary Value Problems,\nRech. Math. Appl. 22,\nMasson, Paris, 1992."},{"key":"2023033111222112637_j_cmam-2020-0030_ref_019","doi-asserted-by":"crossref","unstructured":"T.  Gudi, N.  Nataraj and K.  Porwal,\nAn interior penalty method for distributed optimal control problems governed by the biharmonic operator,\nComput. Math. Appl. 68 (2014), no. 12, 2205\u20132221.","DOI":"10.1016\/j.camwa.2014.08.012"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_020","doi-asserted-by":"crossref","unstructured":"M. D.  Gunzburger, L.  Hou and T. P.  Svobodny,\nAnalysis and finite element approximation of optimal control problems for the stationary Navier\u2013Stokes equations with distributed and Neumann controls,\nMath. Comp. 57 (1991), no. 195, 123\u2013151.","DOI":"10.1090\/S0025-5718-1991-1079020-5"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_021","doi-asserted-by":"crossref","unstructured":"M. D.  Gunzburger and L. S.  Hou,\nFinite-dimensional approximation of a class of constrained nonlinear optimal control problems,\nSIAM J. Control Optim. 34 (1996), no. 3, 1001\u20131043.","DOI":"10.1137\/S0363012994262361"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_022","doi-asserted-by":"crossref","unstructured":"M. D.  Gunzburger, L. S.  Hou and T. P.  Svobodny,\nAnalysis and finite element approximation of optimal control problems for the stationary Navier\u2013Stokes equations with Dirichlet controls,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 25 (1991), no. 6, 711\u2013748.","DOI":"10.1051\/m2an\/1991250607111"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_023","doi-asserted-by":"crossref","unstructured":"L. S.  Hou and J. C.  Turner,\nFinite element approximation of optimal control problems for the von K\u00e1rm\u00e1n equations,\nNumer. Methods Partial Differential Equations 11 (1995), no. 1, 111\u2013125.","DOI":"10.1002\/num.1690110109"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_024","doi-asserted-by":"crossref","unstructured":"G. H.  Knightly,\nAn existence theorem for the von K\u00e1rm\u00e1n equations,\nArch. Ration. Mech. Anal. 27 (1967), 233\u2013242.","DOI":"10.1007\/BF00290614"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_025","doi-asserted-by":"crossref","unstructured":"K.  Krumbiegel and J.  Pfefferer,\nSuperconvergence for Neumann boundary control problems governed by semilinear elliptic equations,\nComput. Optim. Appl. 61 (2015), no. 2, 373\u2013408.","DOI":"10.1007\/s10589-014-9718-0"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_026","doi-asserted-by":"crossref","unstructured":"M.  Li, X.  Guan and S.  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Math. 42 (2016), no. 5, 1031\u20131054.","DOI":"10.1007\/s10444-016-9452-5"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_030","doi-asserted-by":"crossref","unstructured":"G.  Mallik, N.  Nataraj and J.-P.  Raymond,\nError estimates for the numerical approximation of a distributed optimal control problem governed by the von Ka\u0155m\u00e1n equations,\nESAIM Math. Model. Numer. Anal. 52 (2018), no. 3, 1137\u20131172.","DOI":"10.1051\/m2an\/2018023"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_031","doi-asserted-by":"crossref","unstructured":"C.  Meyer and A.  R\u00f6sch,\nSuperconvergence properties of optimal control problems,\nSIAM J. Control Optim. 43 (2004), no. 3, 970\u2013985.","DOI":"10.1137\/S0363012903431608"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_032","doi-asserted-by":"crossref","unstructured":"T.  Miyoshi,\nA mixed finite element method for the solution of the von K\u00e1rm\u00e1n equations,\nNumer. Math. 26 (1976), no. 3, 255\u2013269.","DOI":"10.1007\/BF01395945"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_033","doi-asserted-by":"crossref","unstructured":"A.  Quarteroni,\nHybrid finite element methods for the von K\u00e1rm\u00e1n equations,\nCalcolo 16 (1979), no. 3, 271\u2013288.","DOI":"10.1007\/BF02575930"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_034","doi-asserted-by":"crossref","unstructured":"L.  Reinhart,\nOn the numerical analysis of the von K\u00e1rm\u00e1n equations: mixed finite element approximation and continuation techniques,\nNumer. Math. 39 (1982), no. 3, 371\u2013404.","DOI":"10.1007\/BF01407870"},{"key":"2023033111222112637_j_cmam-2020-0030_ref_035","doi-asserted-by":"crossref","unstructured":"F.  Tr\u00f6ltzsch,\nOptimal Control of Partial Differential Equations. Theory, Methods and Applications,\nGrad. Stud. Math. 112,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/gsm\/112\/07"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/21\/1\/article-p233.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0030\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0030\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T14:34:06Z","timestamp":1680273246000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0030\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,15]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,2,7]]},"published-print":{"date-parts":[[2021,1,1]]}},"alternative-id":["10.1515\/cmam-2020-0030"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2020-0030","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,15]]}}}