An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

Michael Hintermüller; Ronald H.W. Hoppe; Yuri Iliash; Michael Kieweg

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 14, Issue: 3, page 540-560
  • ISSN: 1292-8119

Abstract

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We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

How to cite

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Hintermüller, Michael, et al. "An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2007): 540-560. <http://eudml.org/doc/90882>.

@article{Hintermüller2007,
abstract = { We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method. },
author = {Hintermüller, Michael, Hoppe, Ronald H.W., Iliash, Yuri, Kieweg, Michael},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {A posteriori error analysis; distributed optimal control problems; control constraints; adaptive finite element methods; residual-type a posteriori error estimators; data oscillations; a posteriori error analysis; distributed optimal control problems; adaptive finite element methods; data oscillations; numerical examples},
language = {eng},
month = {11},
number = {3},
pages = {540-560},
publisher = {EDP Sciences},
title = {An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints},
url = {http://eudml.org/doc/90882},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Hintermüller, Michael
AU - Hoppe, Ronald H.W.
AU - Iliash, Yuri
AU - Kieweg, Michael
TI - An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/11//
PB - EDP Sciences
VL - 14
IS - 3
SP - 540
EP - 560
AB - We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.
LA - eng
KW - A posteriori error analysis; distributed optimal control problems; control constraints; adaptive finite element methods; residual-type a posteriori error estimators; data oscillations; a posteriori error analysis; distributed optimal control problems; adaptive finite element methods; data oscillations; numerical examples
UR - http://eudml.org/doc/90882
ER -

References

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