{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T07:28:32Z","timestamp":1775546912872,"version":"3.50.1"},"reference-count":36,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1170672"],"award-info":[{"award-number":["1170672"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,10,1]]},"abstract":"Abstract<\/jats:title>\n We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in \n \n \n \n \n H<\/m:mi>\n \n 1<\/m:mn>\n \/<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u0393<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H^{1\/2}(\\Gamma)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nTo avoid computing the latter norm numerically, we realize it using the \n \n \n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H^{1}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norm of the harmonic extension of the control.\nWe propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier.\nIn the case of convex polygonal domains, optimal error estimates in the \n \n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {H^{1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n <\/m:math>\n \n {L_{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norm are proven.\nWe also consider and analyze the case of control constrained problems.<\/jats:p>","DOI":"10.1515\/cmam-2019-0104","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T07:14:25Z","timestamp":1597734865000},"page":"827-843","source":"Crossref","is-referenced-by-count":9,"title":["A Finite Element Method for Elliptic Dirichlet Boundary Control Problems"],"prefix":"10.1515","volume":"20","author":[{"given":"Michael","family":"Karkulik","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica , Universidad T\u00e9cnica Federico Santa Mar\u00eda , Avenida Espa\u00f1a 1680 , Valpara\u00edso , Chile"}]}],"member":"374","published-online":{"date-parts":[[2020,8,5]]},"reference":[{"key":"2023033110450889239_j_cmam-2019-0104_ref_001","doi-asserted-by":"crossref","unstructured":"T. Apel, M. Mateos, J. Pfefferer and A. R\u00f6sch,\nOn the regularity of the solutions of Dirichlet optimal control problems in polygonal domains,\nSIAM J. Control Optim. 53 (2015), no. 6, 3620\u20133641.","DOI":"10.1137\/140994186"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_002","doi-asserted-by":"crossref","unstructured":"T. Apel, M. Mateos, J. Pfefferer and A. R\u00f6sch,\nError estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes,\nMath. Control Relat. Fields 8 (2018), no. 1, 217\u2013245.","DOI":"10.3934\/mcrf.2018010"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_003","doi-asserted-by":"crossref","unstructured":"T. Apel, S. Nicaise and J. Pfefferer,\nDiscretization of the Poisson equation with non-smooth data and emphasis on non-convex domains,\nNumer. Methods Partial Differential Equations 32 (2016), no. 5, 1433\u20131454.","DOI":"10.1002\/num.22057"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_004","doi-asserted-by":"crossref","unstructured":"F. Ben Belgacem, H. El Fekih and H. Metoui,\nSingular perturbation for the Dirichlet boundary control of elliptic problems,\nM2AN Math. Model. Numer. Anal. 37 (2003), no. 5, 883\u2013850.","DOI":"10.1051\/m2an:2003057"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_005","doi-asserted-by":"crossref","unstructured":"P. Benner and H. Y\u00fccel,\nAdaptive symmetric interior penalty Galerkin method for boundary control problems,\nSIAM J. Numer. Anal. 55 (2017), no. 2, 1101\u20131133.","DOI":"10.1137\/15M1034507"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_006","doi-asserted-by":"crossref","unstructured":"D. Braess,\nFinite Elements. Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed.,\nCambridge University, Cambridge, 2007.","DOI":"10.1017\/CBO9780511618635"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_007","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":"2023033110450889239_j_cmam-2019-0104_ref_008","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nQuasi-interpolation and a posteriori error analysis in finite element methods,\nM2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187\u20131202.","DOI":"10.1051\/m2an:1999140"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_009","doi-asserted-by":"crossref","unstructured":"E. Casas, M. Mateos and J.-P. Raymond,\nPenalization of Dirichlet optimal control problems,\nESAIM Control Optim. Calc. Var. 15 (2009), no. 4, 782\u2013809.","DOI":"10.1051\/cocv:2008049"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_010","doi-asserted-by":"crossref","unstructured":"E. Casas and J.-P. Raymond,\nError estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations,\nSIAM J. Control Optim. 45 (2006), no. 5, 1586\u20131611.","DOI":"10.1137\/050626600"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_011","doi-asserted-by":"crossref","unstructured":"L. Chang, W. Gong and N. Yan,\nWeak boundary penalization for Dirichlet boundary control problems governed by elliptic equations,\nJ. Math. Anal. Appl. 453 (2017), no. 1, 529\u2013557.","DOI":"10.1016\/j.jmaa.2017.04.016"},{"key":"2023033110450889239_j_cmam-2019-0104_ref_012","doi-asserted-by":"crossref","unstructured":"S. Chowdhury, T. Gudi and A. K. Nandakumaran,\nError bounds for a Dirichlet boundary control problem based on energy spaces,\nMath. 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