Dual-weighted goal-oriented adaptive finite elements for optimal control of elliptic variational inequalities∗
Department of Mathematics, Humboldt-Universität zu Berlin, Unter
den Linden 6, 10099
2 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
3 Institute of Mathematics, University of Augsburg, Universitätsstraße 14, 86152 Augsburg, Germany
Revised: 24 June 2013
A dual-weighted residual approach for goal-oriented adaptive finite elements for a class of optimal control problems for elliptic variational inequalities is studied. The development is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Also, a priori bounds for C-stationary points and associated multipliers are derived. Details on the numerical realization of the adaptive concept are provided and a report on numerical tests including the critical cases of biactivity are presented.
Mathematics Subject Classification: 49M25 / 65K15 / 90C33
Key words: Adaptive finite element method / C-stationarity / goal-oriented error estimation / mathematical programming with equilibrium constraints / optimal control of variational inequalities
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”, 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”, and support through a J. Tinsely Oden Fellowship at the Institute for Computational Engineering and Sciences (ICES) at UT Austin, Texas, USA. R.H. has been supported by NSF-DMS 1115658, by the DFG-SPP 1253 and DFG-SPP 1506, by the BMBF projects “FROPT” and “MeFreSim”, and by the ESF within the Research Networking Programme “OPTPDE”. C.L. was supported by the DFG through SPP 1253 “Optimization with Partial Differential Equations”.
© EDP Sciences, SMAI, 2014