{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T11:50:20Z","timestamp":1784116220891,"version":"3.55.0"},"reference-count":47,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-19-13035"],"award-info":[{"award-number":["DMS-19-13035"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We investigate a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mi>P<\/m:mi>\n                            <m:mn>1<\/m:mn>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2021-0106_ineq_0001.png\"\/>\n                        <jats:tex-math>P_{1}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective\/convective and reactive terms.\nThe convergence of this method can be established for general polygonal\/polyhedral domains that are not necessarily convex.\nThe discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.\n                  <\/jats:p>","DOI":"10.1515\/cmam-2021-0106","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T06:57:29Z","timestamp":1630393049000},"page":"777-790","source":"Crossref","is-referenced-by-count":5,"title":["A \ud835\udc43\n                    <sub>1<\/sub>\n                    Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints"],"prefix":"10.1515","volume":"21","author":[{"given":"Susanne C.","family":"Brenner","sequence":"first","affiliation":[{"name":"Department of Mathematics and Center for Computation and Technology , Louisiana State University , Baton Rouge , LA 70803 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sijing","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Connecticut , Storrs , CT 06269 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Li-Yeng","family":"Sung","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Center for Computation and Technology , Louisiana State University , Baton Rouge , LA 70803 , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2021,6,23]]},"reference":[{"key":"2026071112004718196_j_cmam-2021-0106_ref_001","unstructured":"R. A. Adams and J. J. F. Fournier,\nSobolev Spaces, 2nd ed.,\nAcademic Press, Amsterdam, 2003."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_002","doi-asserted-by":"crossref","unstructured":"S. Agmon, A. Douglis and L. Nirenberg,\nEstimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,\nComm. Pure Appl. Math. 12 (1959), 623\u2013727.","DOI":"10.1002\/cpa.3160120405"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_003","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, R. B. Kellogg and J. Pitk\u00e4ranta,\nDirect and inverse error estimates for finite elements with mesh refinements,\nNumer. Math. 33 (1979), no. 4, 447\u2013471.","DOI":"10.1007\/BF01399326"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_004","doi-asserted-by":"crossref","unstructured":"M. Bergounioux, K. Ito and K. Kunisch,\nPrimal-dual strategy for constrained optimal control problems,\nSIAM J. Control Optim. 37 (1999), no. 4, 1176\u20131194.","DOI":"10.1137\/S0363012997328609"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_005","doi-asserted-by":"crossref","unstructured":"M. Bergounioux and K. Kunisch,\nPrimal-dual strategy for state-constrained optimal control problems,\nComput. Optim. Appl. 22 (2002), no. 2, 193\u2013224.","DOI":"10.1023\/A:1015489608037"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_006","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, C. B. Davis and L.-Y. Sung,\nA partition of unity method for a class of fourth order elliptic variational inequalities,\nComput. Methods Appl. Mech. Engrg. 276 (2014), 612\u2013626.","DOI":"10.1016\/j.cma.2014.04.004"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_007","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, J. Gedicke and L.-Y. Sung,\n\n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     C^{0}\n                  \n                interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints,\nSIAM J. Numer. Anal. 56 (2018), no. 3, 1758\u20131785.","DOI":"10.1137\/17M1140649"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_008","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, T. Gudi, K. Porwal and L.-Y. Sung,\nA Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints,\nESAIM Control Optim. Calc. Var. 24 (2018), no. 3, 1181\u20131206.","DOI":"10.1051\/cocv\/2017031"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_009","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, M. Oh, S. Pollock, K. Porwal, M. Schedensack and N. S. Sharma,\nA \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     C^{0}\n                  \n                interior penalty method for elliptic distributed optimal control problems in three dimensions with pointwise state constraints,\nTopics in Numerical Partial Differential Equations and Scientific Computing,\nIMA Vol. Math. Appl. 160,\nSpringer, New York (2016), 1\u201322.","DOI":"10.1007\/978-1-4939-6399-7_1"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_010","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, M. Oh and L.-Y. Sung,\n\n                  \n                     \n                        \n                           P\n                           1\n                        \n                     \n                     \n                     P_{1}\n                  \n                finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions,\nRINAM 8 (2020), Article ID 100090.","DOI":"10.1016\/j.rinam.2019.100090"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_011","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_012","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L.-Y. Sung,\nA new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints,\nSIAM J. Control Optim. 55 (2017), no. 4, 2289\u20132304.","DOI":"10.1137\/16M1088090"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_013","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, L.-Y. Sung and J. Gedicke,\n\n                  \n                     \n                        \n                           P\n                           1\n                        \n                     \n                     \n                     P_{1}\n                  \n                finite element methods for an elliptic optimal control problem with pointwise state constraints,\nIMA J. Numer. Anal. 40 (2020), no. 1, 1\u201328.","DOI":"10.1093\/imanum\/dry071"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_014","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, L.-Y. Sung and Z. Tan,\nA cubic \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     C^{\\textbf{0}}\n                  \n                interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints,\nRINAM 7 (2020), Article ID 100119.","DOI":"10.1016\/j.rinam.2020.100119"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_015","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, L.-Y. Sung and Z. Tan,\nA \n                  \n                     \n                        \n                           C\n                           1\n                        \n                     \n                     \n                     C^{1}\n                  \n                virtual element method for an elliptic distributed optimal control problem with pointwise state constraints, preprint (2021).","DOI":"10.1142\/S0218202521500640"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_016","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, L.-Y. Sung and Y. Zhang,\nA quadratic \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     C^{0}\n                  \n                interior penalty method for an elliptic optimal control problem with state constraints,\nRecent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations,\nIMA Vol. Math. Appl. 157,\nSpringer, Cham (2014), 97\u2013132.","DOI":"10.1007\/978-3-319-01818-8_4"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_017","doi-asserted-by":"crossref","unstructured":"S. C. Brenner, L.-Y. Sung and Y. 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Hamburg 36 (1971), 140\u2013149.","DOI":"10.1007\/BF02995917"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_027","doi-asserted-by":"crossref","unstructured":"J. Frehse,\nOn the regularity of the solution of the biharmonic variational inequality,\nManuscripta Math. 9 (1973), 91\u2013103.","DOI":"10.1007\/BF01320669"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_028","unstructured":"R. Fritzsch and P. Oswald,\nZur optimalen Gitterwahl bei Finite-Elemente-Approximationen,\nWiss. Z. Tech. Univ. Dresden 37 (1988), no. 3, 155\u2013158."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_029","doi-asserted-by":"crossref","unstructured":"W. Gong and N. Yan,\nA mixed finite element scheme for optimal control problems with pointwise state constraints,\nJ. Sci. Comput. 46 (2011), no. 2, 182\u2013203.","DOI":"10.1007\/s10915-010-9392-z"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_030","unstructured":"P. Grisvard,\nElliptic Problems in Nonsmooth Domains,\nPitman, Boston, 1985."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_031","doi-asserted-by":"crossref","unstructured":"M. Heinkenschloss and D. Leykekhman,\nLocal error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems,\nSIAM J. Numer. Anal. 47 (2010), no. 6, 4607\u20134638.","DOI":"10.1137\/090759902"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_032","doi-asserted-by":"crossref","unstructured":"M. Hinterm\u00fcller, K. Ito and K. Kunisch,\nThe primal-dual active set strategy as a semismooth Newton method,\nSIAM J. Optim. 13 (2003), no. 3, 865\u2013888.","DOI":"10.1137\/S1052623401383558"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_033","unstructured":"M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich,\nOptimization with PDE Constraints,\nSpringer, New York, 2009."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_034","doi-asserted-by":"crossref","unstructured":"K. Ito and K. Kunisch,\nLagrange Multiplier Approach to Variational Problems and Applications,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2008.","DOI":"10.1137\/1.9780898718614"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_035","doi-asserted-by":"crossref","unstructured":"D. Kinderlehrer and G. Stampacchia,\nAn Introduction to Variational Inequalities and Their Applications,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000.","DOI":"10.1137\/1.9780898719451"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_036","doi-asserted-by":"crossref","unstructured":"D. Leykekhman and M. Heinkenschloss,\nLocal error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems,\nSIAM J. Numer. Anal. 50 (2012), no. 4, 2012\u20132038.","DOI":"10.1137\/110826953"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_037","unstructured":"W. Liu, W. Gong and N. Yan,\nA new finite element approximation of a state-constrained optimal control problem,\nJ. Comput. Math. 27 (2009), no. 1, 97\u2013114."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_038","doi-asserted-by":"crossref","unstructured":"V. Maz\u2019ya and J. Rossmann,\nElliptic Equations in Polyhedral Domains,\nAmerican Mathematical Society, Providence, 2010.","DOI":"10.1090\/surv\/162"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_039","unstructured":"C. Meyer,\nError estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints,\nControl Cybernet. 37 (2008), no. 1, 51\u201383."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_040","doi-asserted-by":"crossref","unstructured":"S. A. Nazarov and B. A. Plamenevsky,\nElliptic Problems in Domains with Piecewise Smooth Boundaries,\nDe Gruyter, Berlin, 1994.","DOI":"10.1515\/9783110848915"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_041","doi-asserted-by":"crossref","unstructured":"I. Neitzel, J. Pfefferer and A. R\u00f6sch,\nFinite element discretization of state-constrained elliptic optimal control problems with semilinear state equation,\nSIAM J. Control Optim. 53 (2015), no. 2, 874\u2013904.","DOI":"10.1137\/140960645"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_042","unstructured":"P.-A. Raviart,\nThe use of numerical integration in finite element methods for solving parabolic equations,\nTopics in Numerical Analysis,\nAcademic Press, London (1973), 233\u2013264."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_043","unstructured":"W. Rudin,\nReal and Complex Analysis,\nMcGraw-Hill, New York, 1966."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_044","doi-asserted-by":"crossref","unstructured":"A. H. Schatz,\nAn observation concerning Ritz\u2013Galerkin methods with indefinite bilinear forms,\nMath. Comp. 28 (1974), 959\u2013962.","DOI":"10.1090\/S0025-5718-1974-0373326-0"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_045","doi-asserted-by":"crossref","unstructured":"A. H. Schatz and L. B. Wahlbin,\nInterior maximum norm estimates for finite element methods,\nMath. Comp. 31 (1977), no. 138, 414\u2013442.","DOI":"10.1090\/S0025-5718-1977-0431753-X"},{"key":"2026071112004718196_j_cmam-2021-0106_ref_046","unstructured":"L. Schwartz,\nTh\u00e9orie des Distributions,\nHermann, Paris, 1966."},{"key":"2026071112004718196_j_cmam-2021-0106_ref_047","unstructured":"V. Thom\u00e9e,\nGalerkin Finite Element Methods for Parabolic Problems, 2nd ed.,\nSpringer, Berlin, 2006."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2021-0106\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2021-0106\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T12:00:59Z","timestamp":1783771259000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2021-0106\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,23]]},"references-count":47,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,6,23]]},"published-print":{"date-parts":[[2021,10,1]]}},"alternative-id":["10.1515\/cmam-2021-0106"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2021-0106","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,23]]}}}