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

top
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

top

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

top
  1. M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000).  
  2. I. Babuska and W. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.15 (1978) 736–754.  
  3. I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001).  
  4. W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich, Birkhäuser, Basel (2003).  
  5. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput.44 (1985) 283–301.  
  6. R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM J. Control Optim.39 (2000) 113–132.  
  7. M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau-Yosida based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim.11 (2000) 495–521.  
  8. P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math.97 (2004) 219–268.  
  9. C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM. Math. Comput.71 (2002) 945–969.  
  10. C. Carstensen and R.H.W. Hoppe, Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations. J. Numer. Math.13 (2005) 19–32.  
  11. C. Carstensen and R.H.W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method. Math. Comp.75 (2006) 1033–1042.  
  12. C. Carstensen and R.H.W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method. Numer. Math.103 (2006) 251–266.  
  13. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal.33 (1996) 1106–1124.  
  14. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations. Cambridge University Press, Cambridge (1995).  
  15. H.O. Fattorini, Infinite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999).  
  16. M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Quart. Appl. Math.LXI (2003) 131–161.  
  17. J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York (1993).  
  18. R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér.30 (1996) 237–263.  
  19. R.H.W. Hoppe and B. Wohlmuth, Hierarchical basis error estimators for Raviart-Thomas discretizations of arbitrary order, in Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Error Estimates, M. Krizek, P. Neittaanmäki and R. Steinberg Eds., Marcel Dekker, New York (1998) 155–167.  
  20. R. Li, W. Liu H. Ma and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim.41 (2002) 1321–1349.  
  21. X.J. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston-Basel-Berlin (1995).  
  22. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin-Heidelberg-New York (1971).  
  23. W. Liu and N. Yan, A posteriori error estimates for distributed optimal control problems. Adv. Comp. Math.15 (2001) 285–309.  
  24. W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems. Preprint, Institute of Mathematics and Statistics, University of Kent, Canterbury (2003).  
  25. P. Morin, R.H. Nochetto and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal.38 (2000) 466–488.  
  26. P. Neittaanmäki and S. Repin, Reliable methods for mathematical modelling. Error control and a posteriori estimates. Elsevier, New York (2004).  
  27. R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York, Stuttgart (1996).  
  28. O. Zienkiewicz and J. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. J. Numer. Meth. Eng.28 (1987) 28–39.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.