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A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems
1*Dept. of Math., University of Houston, Houston TX 77204-3008, U.S.A.
†Inst. of Math., University of Augsburg, D-86159 Augsburg, Germany
‡Inst. of Math., University of Augsburg, D-86159 Augsburg, Germany
Citation Information: Journal of Numerical Mathematics. Volume 17, Issue 3, Pages 219–244, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: 10.1515/JNUM.2009.012, October 2009
- Published Online:
We provide an a posteriori error analysis of finite element approximations of pointwise state constrained distributed optimal control problems for second order elliptic boundary value problems. In particular, we derive a residual-type a posteriori error estimator and prove its efficiency and reliability up to oscillations in the data of the problem and a consistency error term. In contrast to the case of pointwise control constraints, the analysis is more complicated, since the multipliers associated with the state constraints live in measure spaces. The analysis essentially makes use of appropriate regularizations of the multipliers both in the continuous and in the discrete regime. Numerical examples are given to illustrate the performance of the error estimator.