American Institute of Mathematical Sciences

March  2018, 8(1): 217-245. doi: 10.3934/mcrf.2018010

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

 1 Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, 85577 Neubiberg, Germany 2 Departamento de Matemáticas, Universidad de Oviedo, 33203 Gijón, Spain 3 Lehrstuhl für Optimalsteuerung, Technische Universität München, 85748 Garching bei München, Germany 4 Fakultät für Mathematik, Universtät Duisburg-Essen, D-45127 Essen, Germany

* Corresponding author: Mariano Mateos

Received  April 2017 Revised  September 2017 Published  January 2018

Fund Project: The project was supported by DFG through the International Research Training Group IGDK 1754 Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures.
The second author was partially supported by the Spanish Ministerio Español de Economía y Competitividad under research projects MTM2014-57531-P and MTM2017-83185-P.

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergent meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in nonconvex domains are provided.

Citation: Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch. Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Mathematical Control & Related Fields, 2018, 8 (1) : 217-245. doi: 10.3934/mcrf.2018010
References:

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References:
Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.
Convergence rates depending on the maximal interior angle in the unconstrained case
Convergence rates depending on the maximal interior angle in the constrained case
Family of quasi-uniform meshes which is not $O(h^2)$-irregular
Family of quasi-uniform $O(h^2)$-irregular meshes
Unconstrained problems. Experimental orders of convergence vs biggest angle.
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