Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T16:01:40.203Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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.

Keywords

A posteriori error analysisdistributed optimal control problemscontrol constraintsadaptive finite element methodsresidual-type a posteriori error estimatorsdata oscillations
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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000).
Babuska, I. and Rheinboldt, W., Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. CrossRef
I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001).
W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich, Birkhäuser, Basel (2003).
Bank, R.E. and Weiser, A., Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283–301. CrossRef
Becker, R., Kapp, H. and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM J. Control Optim. 39 (2000) 113–132. CrossRef
Bergounioux, M., Haddou, M., Hintermüller, M. and Kunisch, K., 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. CrossRef
Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. CrossRef
Carstensen, C. and Bartels, S., 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. CrossRef
Carstensen, C. and Hoppe, R.H.W., Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations. J. Numer. Math. 13 (2005) 19–32. CrossRef
Carstensen, C. and Hoppe, R.H.W., Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75 (2006) 1033–1042. CrossRef
Carstensen, C. and Hoppe, R.H.W., Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103 (2006) 251–266. CrossRef
Dörfler, W., A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. CrossRef
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations. Cambridge University Press, Cambridge (1995).
H.O. Fattorini, Infinite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999).
M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Quart. Appl. Math. LXI (2003) 131–161.
J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York (1993).
Hoppe, R.H.W. and Wohlmuth, B., Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 237–263. CrossRef
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.
Li, R., Liu, W. Ma, H. and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41 (2002) 1321–1349. CrossRef
X.J. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston-Basel-Berlin (1995).
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin-Heidelberg-New York (1971).
Liu, W. and Yan, N., A posteriori error estimates for distributed optimal control problems. Adv. Comp. Math. 15 (2001) 285–309. CrossRef
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).
Morin, P., Nochetto, R.H. and Siebert, K.G., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466–488. CrossRef
P. Neittaanmäki and S. Repin, Reliable methods for mathematical modelling. Error control and a posteriori estimates. Elsevier, New York (2004).
R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York, Stuttgart (1996).
Zienkiewicz, O. and Zhu, J., A simple error estimator and adaptive procedure for practical engineering analysis. J. Numer. Meth. Eng. 28 (1987) 28–39.