Abstract
We develop an adaptive finite element method for a class of distributed optimal control problems with control constraints. The method is based on a residual-type a posteriori error estimator and incorporates data oscillations. The analysis is carried out for conforming P1 approximations of the state and the co-state and elementwise constant approximations of the control and the co-control. We prove convergence of the error in the state, the co-state, the control, and the co-control. Under some additional non-degeneracy assumptions on the continuous and the discrete problems, we then show that an error reduction property holds true at least asymptotically. The analysis uses the reliability and the discrete local efficiency of the a posteriori estimator as well as quasi-orthogonality properties as essential tools. Numerical results illustrate the performance of the adaptive algorithm.
The second author has been partially supported by the NSF under Grant No. DMS-0411403 and Grant No. DMS-0511611. The fourth author acknowledges the support by the elite graduate school TopMath.
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References
M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester, 2000.
I. Babuska and W. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978), 736–754.
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.
R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985), 283–301.
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.
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.
P. Binev, W. Dahmen, and R. DeVore, Adaptive Finite Element Methods with Convergence Rates. Numer. Math. 97, (2004), 219–268.
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.
C. Carstensen and R.H.W. Hoppe, Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. (2006) (in press).
C. Carstensen and R.H.W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. (2006) (in press).
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.
W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996), 1106–1124.
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. Quarterly of Applied Mathematics LXI (2003), 131–161.
M. Hintermüller, R.H.W. Hoppe, Y. Iliash, and M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. to appear in ESAIM (2006).
J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York, 1993.
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.
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin-Heidelberg-New York, 1971.
W. Liu and N. Yan, A posteriori error estimates for distributed optimal control problems. Adv. Comp. Math. 15 (2001), 285–309.
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).
X.J. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser, Boston-Basel-Berlin, 1995.
P. Morin, R.H. Nochetto, and K.G. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000), 466–488.
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.
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Gaevskaya, A., Iliash, Y., Kieweg, M., Hoppe, R.H.W. (2007). Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7721-2_3
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DOI: https://doi.org/10.1007/978-3-7643-7721-2_3
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