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Error bounds for a Dirichlet boundary control problem based on energy spaces
Authors: Sudipto Chowdhury, Thirupathi Gudi and A. K. Nandakumaran
Journal: Math. Comp. 86 (2017), 1103-1126
MSC (2010): Primary 65N30, 65N15, 65N12, 65K10
DOI: https://doi.org/10.1090/mcom/3125
Published electronically: June 20, 2016
MathSciNet review: 3614013
Abstract | References | Similar Articles | Additional Information
Abstract: In this article, an alternative energy-space based approach is proposed for the Dirichlet boundary control problem and then a finite-element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the $L_2$-norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help of an auxiliary problem. The theoretical results are illustrated by the numerical experiments.
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Eduardo Casas and Mariano Mateos, Error estimates for the numerical approximation of Neumann control problems, Comput. Optim. Appl. 39 (2008), no. 3, 265–295. MR 2396868 (2009a:49055), DOI https://doi.org/10.1007/s10589-007-9056-6
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Tunç Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numér. 13 (1979), no. 4, 313–328 (English, with French summary). MR 555382 (80j:93060)
Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683 (86m:35044)
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M. D. Gunzburger, L. S. Hou, and Th. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 711–748 (English, with French summary). MR 1135991 (92k:49061)
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M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), no. 1, 45–61. MR 2122182 (2005i:49034), DOI https://doi.org/10.1007/s10589-005-4559-5
Michael Hintermüller, Ronald H. W. Hoppe, Yuri Iliash, and Michael Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM Control Optim. Calc. Var. 14 (2008), no. 3, 540–560. MR 2434065 (2009e:65175), DOI https://doi.org/10.1051/cocv%3A2007057
Ruo Li, Wenbin Liu, Heping Ma, and Tao Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41 (2002), no. 5, 1321–1349. MR 1971952 (2004a:49036), DOI https://doi.org/10.1137/S0363012901389342
Wenbin Liu and Ningning Yan, A posteriori error estimates for convex boundary control problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 73–99. MR 1860717 (2002j:49040), DOI https://doi.org/10.1137/S0036142999352187
C. Meyer and A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim. 43 (2004), no. 3, 970–985 (electronic). MR 2114385 (2005i:49024), DOI https://doi.org/10.1137/S0363012903431608
C. Ortner and W. Wollner, A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state, Numer. Math. 118 (2011), no. 3, 587–600. MR 2810808 (2012g:65105), DOI https://doi.org/10.1007/s00211-011-0360-9
Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N15, 65N12, 65K10
Retrieve articles in all journals with MSC (2010): 65N30, 65N15, 65N12, 65K10
Sudipto Chowdhury
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
Email: sudipto10@math.iisc.ernet.in
Thirupathi Gudi
Email: gudi@math.iisc.ernet.in
A. K. Nandakumaran
Email: nands@math.iisc.ernet.in
Keywords: Optimal control, Dirichlet control, finite element, optimal error estimate, adaptive finite element, a posteriori estimates
Received by editor(s): May 19, 2015
Received by editor(s) in revised form: August 27, 2015, and October 7, 2015
Article copyright: © Copyright 2016 American Mathematical Society
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