A posteriori error estimates for adaptive finite element discretizations of boundary control problems

Journal of Numerical Mathematics

Volumn 14, Issue 1, 2006, Pages 57-82

  Hoppe, R H W ^a,b   Iliash, Y ^a   Iyyunni, C ^b   Sweilam, N H ^c  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b University of Houston   (United States)

^c CAIRO UNIVERSITY   (Egypt)

Author keywords

A posteriori error analysis; Adaptive finite element methods; Boundary control problems; Control constraints; Data oscillations; Residual type aposteriori error estimators

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; CONSTRAINT THEORY; FINITE ELEMENT METHOD; NUMERICAL ANALYSIS;

A POSTERIORI ERROR ANALYSIS; ADAPTIVE FINITE ELEMENT METHODS; BOUNDARY CONTROL PROBLEMS; CONTROL CONSTRAINTS; DATA OSCILLATIONS; RESIDUAL-TYPE APOSTERIORI ERROR ESTIMATORS;

ERROR ANALYSIS;

EID: 33744492592     PISSN: 15702820     EISSN: None     Source Type: Journal    
DOI: 10.1163/156939506776382139     Document Type: Article

References (29)

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. (1978) 15, 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. Comp. (1985) 44, 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. (2000) 39, 113-132.
7. M. Bergounioux, M. Haddou, M. Hintermüller, and K. Kunisch, A comparison of a MoreauYosida based active set strategy and interior point methods for constrained optimal control problems. SIAMJ. Optim. (2000) 11, 495-521.
8. P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates. Numer. Math. (2004) 97, 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. Comp. (2002) 71, 945-969.
10. C. Carstensen and R. H. W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. (2004) (in press).
11. C. Carstensen and R. H. W. Hoppe, R.H.W. Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. (2005) (to appear).
12. 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. (2005) 13, 19-32.
13. W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAMJ. Numer. Anal. (1996) 33, No. 3, 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. Quarterly Appl. Math. (2003) LXI, 131 -161.
17. M. Hintermüller, R.H.W. Hoppe, Yu. Iliash, and M.Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. (2005) (submitted).
18. J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York, 1993.
19. 2 AN, Modél. Math. Anal. Numér. (1996) 30, 237 - 263.
20. 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 (Eds. M. Krizek, P. Neittaanmäki, and R. Steinberg), Marcel Dekker, New York, 1998, pp. 155 -167.
21. R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. (2002) 41, 1321 -1349.
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. (2001) 15, 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. X. J. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston-Basel-Berlin, 1995.
26. P. Morin, R. H. Nochetto, and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. (2000) 38, No. 2, 466-488.
27. P. Neittaanmäki and S. Repin, Reliable Methods for Mathematical Modelling. Error Control and A Posteriori Estimates. Elsevier, New York, 2004.
28. R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York-Stuttgart, 1996.
29. O. Zienkiewicz and J. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. J. Numer. Meth. Engrg. (1987) 28, 28-39.