Computational Optimization and Applications manuscript No. (will be inserted by the editor) A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints Olaf Benedix ⋆ , Boris Vexler Lehrstuhl f¨ ur Mathematische Optimierung, Technische Universit¨at M¨ unchen, Fakult¨atf¨ ur Mathematik, Boltzmannstraße 3, Garching b. M¨ unchen, Germany, e-mail: {benedix,vexler}@ma.tum.de Received: date / Revised version: date Abstract In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are con- sidered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method. Key words optimal control, state constraints, semilinear equations, finite elements, a posteriori error estimation 1 Introduction In this paper we consider optimal control problems governed by semilinear elliptic partial differential equations and subject to inequality constraints on the state variable, so called state constraints, formulated as follows:      minimize J (q,u),q ∈ Q, u ∈ V, u = S(q), u a (x) ≤ u(x) ≤ u b (x) for x ∈ ¯ Ω. (1.1) Here, the pair (q,u) consists of the control variable q and the state variable u from the corresponding function spaces Q and V to be specified later. ⋆ The author’s research was supported by the Austrian Science Fund FWF, project P18971-N18 ”Numerical analysis and discretization strategies for optimal control problems with singularities”