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 Ω. To solve this problem numerically, it is usually necessary to approximate Ω by a (typically polygonal) new domain Ωh. The difference between the solutions of both i...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2011 |
| País: | España |
| Recursos: | Universidad de Cantabria (UC) |
| Repositório: | UCrea Repositorio Abierto de la Universidad de Cantabria |
| Idioma: | inglês |
| OAI Identifier: | oai:repositorio.unican.es:10902/1638 |
| Acesso em linha: | http://hdl.handle.net/10902/1638 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Dirichlet control Error estimates Semilinear elliptic equations Finite elements |
| Resumo: | In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain Ω. To solve this problem numerically, it is usually necessary to approximate Ω by a (typically polygonal) new domain Ωh. The difference between the solutions of both infinite-dimensional control problems, one formulated in Ω and the second in Ωh, 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. G¨unther, and M. Hinze, SIAM J. Control Optim., 48 (2009), pp. 2798–2819], the numerical approximation of the problem defined in Ω was considered. The authors used a finite element method such that Ωh 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 Ω to Ωh. |
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