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

Thomas Apel; Mariano Mateos; Johannes Pfefferer; Arnd Rösch

doi:10.3934/mcrf.2018010

Article Contents

2018, Volume 8, Issue 1: 217-245. Doi: 10.3934/mcrf.2018010

This issue Previous Article Error analysis for global minima of semilinear optimal control problems Next Article Optimal control of a non-smooth semilinear elliptic equation

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
* Corresponding author: Mariano Mateos

Received: April 2017

Revised: September 2017

Published: January 2018

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.

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Abstract

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.

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Mathematics Subject Classification: 65N30, 65N15, 49M05, 49M25.

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Figure 6. Constrained problems. Experimental orders of convergence vs biggest angle. Left: generic case. Right: worst case.

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Figure 1. Convergence rates depending on the maximal interior angle in the unconstrained case

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Figure 2. Convergence rates depending on the maximal interior angle in the constrained case

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Figure 3. Family of quasi-uniform meshes which is not $O(h^2)$-irregular

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Figure 4. Family of quasi-uniform $O(h^2)$-irregular meshes

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Figure 5. Unconstrained problems. Experimental orders of convergence vs biggest angle.

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