Abstract
In this paper, we study adaptive finite element approximation schemes for a constrained optimal control problem. We derive the equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ainsworth, M., Oden, J.T.: A posteriori error estimators in finite element analysis. Comput. Methods Appl. Mech. Eng. 142, 1–88 (1997)
Chang, Y.Z., Yang, D.P.: Superconvergence analysis of finite element methods for optimal control problems of the stationary Benard type. J. Comput. Math. 26, 660–676 (2008)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation. In: Iserles, A. (ed.) Acta Numer, pp. 1–102. Cambridge University Press, Cambridge (2001)
Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39, 113–132 (2000)
Heinkenschloss, K., Vicente, L.N.: Analysis of inexact trust-region SQP algorithms. SIAM J. Optim. 12, 283–302 (2001)
Huang, Y.Q., Li, R., Liu, W.B., Yan, N.N.: Adaptive multi-mesh finite element approximation for constrained optimal control. Submitted to SIAM Optim. Control (first in 2004)
Kelley, C.T., Sachs, E.W.: A trust region method for parabolic boundary control problems. SIAM J. Optim. 9, 1064–1091 (1999)
Kufner, A., John, O., Fucik, S.: Function Spaces. Nordhoff, Leiden (1977)
Li, R.: On multi-mesh h-adaptive algorithm. JSC 24, 321–341 (2005)
Li, R., Liu, W.B., Ma, H.P., Tang, T.: Adaptive finite element approximation of elliptic optimal control. SIAM J. Control. Optim. 41, 1321–1349 (2002)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Liu, W.B.: Adaptive multi-meshes in finite element approximation of optimal control. Contemp. Math. 383, 113–132 (2005)
Liu, W.B., Gong, W., Yan, N.N.: A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27, 97–114 (2009)
Liu, W.B., Yan, N.N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comput. Math. 15(1–4), 285–309 (2001)
Sachs, E., Volkwein, S.: Augmented Lagrange-SQP methods with Lipschitz continuous Lagrange multiplier updates. SIAM J. Numer. Anal. 40, 233–253 (2002)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Verfurth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989)
Verfurth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, London (1996)
Volkwein, S.: Affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods. SIAM J. Control. Optim. 41, 875–899 (2002)
Yang, D.P., Chang, Y.Z., Liu, W.B.: A priori error estimate and superconvergence analysis for an optimal control problem of bilinear type. J. Comput. Math. 26, 471–487 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Basic Research Program of P.R. China under the grant 2005CB321703.
Rights and permissions
About this article
Cite this article
Ge, L., Liu, W. & Yang, D. Adaptive Finite Element Approximation for a Constrained Optimal Control Problem via Multi-meshes. J Sci Comput 41, 238 (2009). https://doi.org/10.1007/s10915-009-9296-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-009-9296-y