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

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

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 540-560.

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.

[1] M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000). | MR | Zbl

[2] I. Babuska and W. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | MR | Zbl

[3] I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001). | MR

[4] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich, Birkhäuser, Basel (2003). | MR | Zbl

[5] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283-301. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[8] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] C. Carstensen and R.H.W. Hoppe, Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103 (2006) 251-266. | MR | Zbl

[13] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[14] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations. Cambridge University Press, Cambridge (1995). | Zbl

[15] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999). | MR | Zbl

[16] M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Quart. Appl. Math. LXI (2003) 131-161. | MR | Zbl

[17] J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York (1993). | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] X.J. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston-Basel-Berlin (1995). | MR | Zbl

[22] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin-Heidelberg-New York (1971). | MR | Zbl

[23] W. Liu and N. Yan, A posteriori error estimates for distributed optimal control problems. Adv. Comp. Math. 15 (2001) 285-309. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[26] P. Neittaanmäki and S. Repin, Reliable methods for mathematical modelling. Error control and a posteriori estimates. Elsevier, New York (2004). | MR | Zbl

[27] R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York, Stuttgart (1996). | Zbl

[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. | MR | Zbl