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

  • * Corresponding author: Mariano Mateos

    * Corresponding author: Mariano Mateos 
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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  • 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.

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

    Citation:

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

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

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

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

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

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

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