Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

Abstract

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established. Abstract. We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established.Key words. Boundary control, semilinear elliptic equation, numerical approximation, error estimates AMS subject classifications. 49J20, 49K20, 49M05, 65K101. Introduction. With this paper, we continue the discussion of error estimates for the numerical approximation of optimal control problems we have started for semilinear elliptic equations and distributed controls in [1]. The case of distributed control is the easiest one with respect to the mathematical analysis. In [1] it was shown that, roughly speaking, the distance between a locally optimal controlū and its numerical approximationū h has the order of the mesh size h in the L 2 -norm and in the L ∞ -norm. This estimate holds for a finite element approximation of the equation by standard piecewise linear elements and piecewise constant control functions.The analysis for boundary controls is more difficult, since the regularity of the state function is lower than that for distributed controls. Moreover, the internal approximation of the domain causes problems. In the general case, we have to approximate the boundary by a polygon. This requires the comparison of the original control that is located at the boundary Γ and the approximate control that is defined on the polygonal boundary Γ h . Moreover, the regularity of elliptic equations in domains with corners needs special care. To simplify the analysis, we assume here that Ω is a polygonal domain of R 2 . Though this makes the things easier, the lower regularity of states in polygonal domains complicates, together with the presence of nonlinearities, the analysis.Another novelty of our paper is the numerical confirmation of the predicted error estimates. We present two examples, where we know the exact solutions. The first one is of linear-quadratic type, while the second one is semilinear. We are able to verify our error estimates quite precisely.Let us mention some further papers related to this subject. The case of linearquadratic elliptic control problems by finite elements was discussed in early papers by Falk [9], Geveci [10] and Malanowski [16], and Arnautu and Neittaanmäki [2], who already proved the optimal error estimate of order h in the L 2 -...

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