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.
Mots clés : a posteriori error analysis, distributed optimal control problems, control constraints, adaptive finite element methods, residual-type a posteriori error estimators, data oscillations
@article{COCV_2008__14_3_540_0, author = {Kieweg, Michael and Iliash, Yuri and Hoppe, Ronald H. W. and Hinterm\"uller, Michael}, title = {An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {540--560}, publisher = {EDP-Sciences}, volume = {14}, number = {3}, year = {2008}, doi = {10.1051/cocv:2007057}, mrnumber = {2434065}, zbl = {1157.65039}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007057/} }
TY - JOUR AU - Kieweg, Michael AU - Iliash, Yuri AU - Hoppe, Ronald H. W. AU - Hintermüller, 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 PY - 2008 SP - 540 EP - 560 VL - 14 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007057/ DO - 10.1051/cocv:2007057 LA - en ID - COCV_2008__14_3_540_0 ER -
%0 Journal Article %A Kieweg, Michael %A Iliash, Yuri %A Hoppe, Ronald H. W. %A Hintermüller, Michael %T An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 540-560 %V 14 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007057/ %R 10.1051/cocv:2007057 %G en %F COCV_2008__14_3_540_0
Kieweg, Michael; Iliash, Yuri; Hoppe, Ronald H. W.; Hintermüller, Michael. 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, Tome 14 (2008) no. 3, pp. 540-560. doi : 10.1051/cocv:2007057. http://www.numdam.org/articles/10.1051/cocv:2007057/
[1] A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000). | MR | Zbl
and ,[2] Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754. | MR | Zbl
and ,[3] The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001). | MR
and ,[4] Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich, Birkhäuser, Basel (2003). | MR | Zbl
and ,[5] Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283-301. | MR | Zbl
and ,[6] Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM J. Control Optim. 39 (2000) 113-132. | MR | Zbl
, and ,[7] 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
, , and ,[8] Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219-268. | MR | Zbl
, and ,[9] 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
and ,[10] Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations. J. Numer. Math. 13 (2005) 19-32. | MR | Zbl
and ,[11] Error reduction and convergence for an adaptive mixed finite element method. Math. Comp. 75 (2006) 1033-1042. | MR | Zbl
and ,[12] Convergence analysis of an adaptive nonconforming finite element method. Numer. Math. 103 (2006) 251-266. | MR | Zbl
and ,[13] A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl
,[14] Computational Differential Equations. Cambridge University Press, Cambridge (1995). | Zbl
, , and ,[15] Infinite Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999). | MR | Zbl
,[16] A primal-dual active set algorithm for bilaterally control constrained optimal control problems. Quart. Appl. Math. LXI (2003) 131-161. | MR | Zbl
,[17] Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York (1993). | Zbl
and ,[18] 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
and ,[19] 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
and ,[20] 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] Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston-Basel-Berlin (1995). | MR | Zbl
and ,[22] Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin-Heidelberg-New York (1971). | MR | Zbl
,[23] A posteriori error estimates for distributed optimal control problems. Adv. Comp. Math. 15 (2001) 285-309. | MR | Zbl
and ,[24] A posteriori error estimates for convex boundary control problems. Preprint, Institute of Mathematics and Statistics, University of Kent, Canterbury (2003). | MR | Zbl
and ,[25] Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000) 466-488. | MR | Zbl
, and ,[26] Reliable methods for mathematical modelling. Error control and a posteriori estimates. Elsevier, New York (2004). | MR | Zbl
and ,[27] A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York, Stuttgart (1996). | Zbl
,[28] A simple error estimator and adaptive procedure for practical engineering analysis. J. Numer. Meth. Eng. 28 (1987) 28-39. | MR | Zbl
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