Numerical Analysis and Scientific Computing Preprint Seria Adaptive finite elements for optimally controlled elliptic variational inequalities of obstacle type

We are concerned with the numerical solution of distributed optimal control problems for second order elliptic variational inequalities by adaptive finite element methods. Both the continuous problem as well as its finite element approximations represent subclasses of Mathematical Programs with Equilibrium Constraints (MPECs) for which the optimality conditions are stated by means of stationarity concepts in function space [30] and in a discrete, finite dimensional setting [50] such as (ε-almost, almost) Cand S-stationarity. With regard to adaptive mesh refinement, in contrast to the work in [28] which adopts a goal oriented dual weighted approach, we consider standard residual-type a posteriori error estimators. The first main result states that for a sequence of discrete C-stationary points there exists a subsequence converging to an almost C-stationary point, provided the associated sequence of nested finite element spaces is limit dense in its continuous counterpart. As the second main result, we prove the reliability and efficiency of the residual-type a posteriori error estimators. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors by heuristically motivated computable quantities and on the approximation of the continuous active, strongly active, and inactive sets by their discrete counterparts. A detailed documentation of numerical results for two representative test examples illustrates the performance of the adaptive approach. Mathematics Subject Classification (2000). Primary 65K15; Secondary 49M99; 65K10; 90C56. S.G. has been partially supported by a grant from the European Science Foundation within the Networking Programme ’OPTPDE’. M.H. acknowledges support by the German Research Fund (DFG) through the Research Center MATHEON Project C28 and C31 and the SPP 1253 ”Optimization with Partial Differential Equations”, and the Austrian Science Fund (FWF) through the START Project Y 305-N18 Interfaces and Free Boundaries and the SFB Project F32 04-N18 ”Mathematical Optimization and Its Applications in Biomedical Sciences”. R.H.W.H. has been supported by the DFG Priority Programs SPP 1253 and SPP 1506, by the NSF grants DMS0914788, DMS-1115658, by the Federal Ministry for Education and Research (BMBF) within the projects ’FROPT’ and ’MeFreSim’, and by the European Science Foundation within the Networking Programme ’OPTPDE’..