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

Abstract: 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.

Otras publicaciones de la misma revista o congreso con autores/as de la Universidad de Cantabria

 Autoría: Casas E., Günther A., Mateos M.,

 Fuente: SIAM Journal on Control and Optimization, 2011, 49(5), 1998-2007

Editorial: Society for Industrial and Applied Mathematics

 Fecha de publicación: 01/01/2011

Nº de páginas: 10

Tipo de publicación: Artículo de Revista

 DOI: 10.1137/100794882

ISSN: 0363-0129,1095-7138

 Proyecto español: MTM2008-04206