A paradox in the approximation of dirichlet control problems in curved domains

In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain O. To solve this problem numerically, it is usually necessary to approximate O by a (typically polygonal) new domain Oh. The difference between the solutions of both i...

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Autores: Casas Rentería, Eduardo|||0000-0002-8364-9416, Günther, Andreas, Mateos Alberdi, Mariano
Tipo de recurso: artículo
Fecha de publicación:2011
País:España
Institución:Universidad de Cantabria (UC)
Repositorio:UCrea Repositorio Abierto de la Universidad de Cantabria
Idioma:inglés
OAI Identifier:oai:repositorio.unican.es:10902/26710
Acceso en línea:https://hdl.handle.net/10902/26710
Access Level:acceso abierto
Palabra clave:Dirichlet control
Error estimates
Semilinear elliptic equations
Finite element
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spelling A paradox in the approximation of dirichlet control problems in curved domainsCasas Rentería, Eduardo|||0000-0002-8364-9416Günther, AndreasMateos Alberdi, MarianoDirichlet controlError estimatesSemilinear elliptic equationsFinite elementIn this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain O. To solve this problem numerically, it is usually necessary to approximate O by a (typically polygonal) new domain Oh. The difference between the solutions of both infinite-dimensional control problems, one formulated in O and the second in Oh, was studied in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780], where an error of order O(h) was proved. In [K. Deckelnick, A. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798-2819], the numerical approximation of the problem defined in O was considered. The authors used a finite element method such that Oh was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order O(h3/2) for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from O to Oh.The first and the third authors were partially supportedby the Spanish Ministry of Science and Innovation under projects MTM2008-04206 and “IngenioMathematica (i-MATH)” CSD2006-00032 (Consolider Ingenio 2010)Society for Industrial and Applied MathematicsUniversidad de Cantabria20112011-01-01journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articlehttps://hdl.handle.net/10902/26710SIAM Journal on Control and Optimization, 2011, 49(5), 1998-2007reponame:UCrea Repositorio Abierto de la Universidad de Cantabriainstname:Universidad de Cantabria (UC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.unican.es:10902/267102026-06-02T12:39:31Z
dc.title.none.fl_str_mv A paradox in the approximation of dirichlet control problems in curved domains
title A paradox in the approximation of dirichlet control problems in curved domains
spellingShingle A paradox in the approximation of dirichlet control problems in curved domains
Casas Rentería, Eduardo|||0000-0002-8364-9416
Dirichlet control
Error estimates
Semilinear elliptic equations
Finite element
title_short A paradox in the approximation of dirichlet control problems in curved domains
title_full A paradox in the approximation of dirichlet control problems in curved domains
title_fullStr A paradox in the approximation of dirichlet control problems in curved domains
title_full_unstemmed A paradox in the approximation of dirichlet control problems in curved domains
title_sort A paradox in the approximation of dirichlet control problems in curved domains
dc.creator.none.fl_str_mv Casas Rentería, Eduardo|||0000-0002-8364-9416
Günther, Andreas
Mateos Alberdi, Mariano
author Casas Rentería, Eduardo|||0000-0002-8364-9416
author_facet Casas Rentería, Eduardo|||0000-0002-8364-9416
Günther, Andreas
Mateos Alberdi, Mariano
author_role author
author2 Günther, Andreas
Mateos Alberdi, Mariano
author2_role author
author
dc.contributor.none.fl_str_mv Universidad de Cantabria
dc.subject.none.fl_str_mv Dirichlet control
Error estimates
Semilinear elliptic equations
Finite element
topic Dirichlet control
Error estimates
Semilinear elliptic equations
Finite element
description In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain O. To solve this problem numerically, it is usually necessary to approximate O by a (typically polygonal) new domain Oh. The difference between the solutions of both infinite-dimensional control problems, one formulated in O and the second in Oh, was studied in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780], where an error of order O(h) was proved. In [K. Deckelnick, A. Günther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798-2819], the numerical approximation of the problem defined in O was considered. The authors used a finite element method such that Oh was the polygon formed by the union of all triangles of the mesh of parameter h. They proved an error of order O(h3/2) for the difference between continuous and discrete optimal controls. Here we show that the estimate obtained in [E. Casas and J. Sokolowski, SIAM J. Control Optim., 48 (2010), pp. 3746-3780] cannot be improved, which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from O to Oh.
publishDate 2011
dc.date.none.fl_str_mv 2011
2011-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/10902/26710
url https://hdl.handle.net/10902/26710
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv SIAM Journal on Control and Optimization, 2011, 49(5), 1998-2007
reponame:UCrea Repositorio Abierto de la Universidad de Cantabria
instname:Universidad de Cantabria (UC)
instname_str Universidad de Cantabria (UC)
reponame_str UCrea Repositorio Abierto de la Universidad de Cantabria
collection UCrea Repositorio Abierto de la Universidad de Cantabria
repository.name.fl_str_mv
repository.mail.fl_str_mv
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