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An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problemswith controlconstraints

Published online by Cambridge University Press:  21 November 2007

Michael Hintermüller ,
Ronald H.W. Hoppe ,
Yuri Iliash  and
Michael Kieweg
Michael Hintermüller
Affiliation:
Institute of Mathematics, University of Graz, 8010 Graz, Austria.
Ronald H.W. Hoppe
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA. Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany; Ronald.Hoppe@math.uni-augsburg.de
Yuri Iliash
Affiliation:
Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany; Ronald.Hoppe@math.uni-augsburg.de
Michael Kieweg
Affiliation:
Institute of Mathematics, University of Augsburg, 86159 Augsburg, Germany; Ronald.Hoppe@math.uni-augsburg.de
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Abstract

We present an a posteriori error analysis of adaptive finiteelement approximations of distributed control problems for secondorder elliptic boundary value problems under bound constraints onthe control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and elementresiduals. Since we do not assume any regularity of the data ofthe problem, the error analysis further invokes data oscillations.We prove reliability and efficiency of the error estimator andprovide a bulk criterion for mesh refinement that also takes intoaccount data oscillations and is realized by a greedy algorithm. Adetailed documentation of numerical results for selected testproblems illustrates the convergence of the adaptive finiteelement method.

Keywords

A posteriori error analysisdistributed optimalcontrol problemscontrol constraintsadaptive finite elementmethodsresidual-type a posteriori error estimatorsdataoscillations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 540 - 560
Copyright
© EDP Sciences, SMAI, 2007

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