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

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 quasiuniform meshes and superconvergence 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 non-convex domains are provided. of the adjoint state has a pole there, we getū ∈ H s (Γ) for all s < 3 2 . This regularity is determined by the larges convex angle and by the kinks due to the constraints.Unfortunately, this is not the whole truth. In exceptional cases, e. g. when the data enjoy certain symmetry, the leading singularity of type r 2/3 may not appear in the adjoint state. Instead, the solution may have a r 4/3 -singularity whose normal derivative has a r 1/3 -singularity which is not flattened by the projection Proj [a,b] . The control is less regular,ū ∈ H s (Γ) for all s < 5 6 . See Example 3.6 in [1]. Hence, dealing with these exceptional cases is not fun but necessary. If in the unconstrained case a stress intensity factor vanishes , i.e., the leading singularity does not occur, then the convergence result is still true, one may only see a better convergence in numerical tests. See Figure 3, right hand side and Remark 4.8. However, in the constrained case, the situation is the opposite. The exceptional case leads to the worstcase estimate. To deal with the "worst-case" and the "usual-case" in an unified way, we introduce in (2.4) some numbers related to the singular exponents.We distinguish two cases for the investigation of the discretization errors. After proving a general result in Section 3 we study the unconstrained case in Section 4 and the constrained case in Section 5. We focus on quasi-uniform meshes and distinguish general meshes and certain superconvergence meshes. In order not to overload the present paper, we postpone the study of graded meshes to [2]. The numerical tests in Section 6 confirm the theoretical results.The study of error estimates for Dirichlet control problems posed on polygonal domains can be traced back to [9], where a control constrained problem governed by a semilinear elliptic equation posed in a convex polygonal domain is studied. An order of convergence of h s is proved for all s < min(1, π/(2ω 1 )), where ω 1 is the largest interior angle, in both the control and the state variable. Later, in [18], it is proven that for unconstrained linear problems posed on convex domains, the state variable exhibits a better convergence property. The corresponding proof is based on a duality argument and estimates for the controls in weaker norms than L 2 (Γ). However, to the best of our knowledge, the argumentation is restricted to unconstrained problems. For the error of t...

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