Abstract
In this paper, we study the adaptive finite element approximation for a constrained optimal control problem with both pointwise and integral control constraints. We first obtain the explicit solutions for the variational inequalities both in the continuous and discrete cases. Then a priori error estimates are established, and furthermore equivalent a posteriori error estimators are derived for both the state and the control approximation, which can be used to guide the mesh refinement for an adaptive multi-mesh finite element scheme. The a posteriori error estimators are implemented and tested with promising numerical results.









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)
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)
Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation. In: Iserles, A. (ed.) Acta Numerica 2001, pp. 1–102. Cambridge University Press, Cambridge (2001)
Bejan, A.: Constructal-theory network of conducting paths for cooling a heat generating volume. Int. J. Heat Mass Transf. 40(4), 779–816 (1997)
Casas, E., Troltzsch, F.: Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 12, 406–431 (2002)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Deckelnick, K., Hinze, M.: Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comp. Math. 29, 1–15 (2011)
Dorfler, W.: A convergent adaptive algorithm for Poisson’s equations. SIAM J. Numer. Anal. 33, 1106–1124 (1996)
Ge, L., Liu, W.B., Yang, D.P.: Adaptive finite element approximation for a constrained optimal control problem via multi-meshes. J. Sci. Comput. 41, 238–255 (2009)
Guo, Z.Y., Cheng, X.G., Xia, Z.Z.: Least dissipation principle of heat transport potential capacity and its application in heat conduction optimization. Chin. Sci. Bull. 48(4), 406–410 (2003)
Heinkenschloss, K., Vicente, L.N.: Analysis fo inexact trust-region SQP algorithms. SIAM J. Optim. 12, 283–302 (2001)
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)
Lee, H., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commun. Comput. Phys. 14, 77–106 (2013)
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)
Li, R.: On Multi-Mesh h-Adaptive Algorithm. J. Sci. Comput. 24, 321–341 (2005)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Liu, W.B., Tiba, D.: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. Numer. Funct. Anal. Optim. 22, 953–972 (2001)
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)
Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs, p. 1. Science Press, Beijing (2008)
Liu, W.B., Yang, D.P., Yuan, L., Ma, C.Q.: Finite element approximations of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48, 1163–1185 (2010)
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2001)
Niu, H., Yang, D.: Finite element analysis of optimal control problem governed by Stokes equations with \(L^2\)-norm state-constraints. J. Comput. Math. 29, 589–604 (2011)
Roos, H., Reibiger, C.: Numerical analysis of a system of singularly perturbed convection-diffusion equations related to optimal control. Numer. Math. Theor. Meth. Appl. 4, 562–575 (2011)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Vallejos, M.: Multigrid methods for elliptic optimal control problems with pointwise state Constraints. Numer. Math. Theor. Meth. Appl. 5, 99–109 (2012)
Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001)
Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989)
Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, London (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Ning Du was supported by the NSFC under the Grant 11371229 and 11201265.
Rights and permissions
About this article
Cite this article
Du, N., Ge, L. & Liu, W. Adaptive Finite Element Approximation for an Elliptic Optimal Control Problem with Both Pointwise and Integral Control Constraints. J Sci Comput 60, 160–183 (2014). https://doi.org/10.1007/s10915-013-9790-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-013-9790-0