## Error bounds for a Dirichlet boundary control problem based on energy spaces

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- by Sudipto Chowdhury, Thirupathi Gudi and A. K. Nandakumaran PDF
- Math. Comp.
**86**(2017), 1103-1126 Request permission

## 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.## References

- Thomas Apel, Johannes Pfefferer, and Arnd Rösch,
*Finite element error estimates for Neumann boundary control problems on graded meshes*, Comput. Optim. Appl.**52**(2012), no. 1, 3–28. MR**2925763**, DOI 10.1007/s10589-011-9427-x - F. B. Belgacem, H. E. Fekih, and H. Metoui,
*Singular perturbation for the Dirichlet boundary control of elliptic problems*, ESAIM:M2AN**37**(2003), 833–850. - Christine Bernardi and Frédéric Hecht,
*Error indicators for the mortar finite element discretization of the Laplace equation*, Math. Comp.**71**(2002), no. 240, 1371–1403. MR**1933036**, DOI 10.1090/S0025-5718-01-01401-6 - Susanne C. Brenner,
*A nonstandard finite element interpolation estimate*, Numer. Funct. Anal. Optim.**20**(1999), no. 3-4, 245–250. MR**1691362**, DOI 10.1080/01630569908816890 - Susanne C. Brenner and Michael Neilan,
*A $\scr C^0$ interior penalty method for a fourth order elliptic singular perturbation problem*, SIAM J. Numer. Anal.**49**(2011), no. 2, 869–892. MR**2792399**, DOI 10.1137/100786988 - Susanne C. Brenner, Thirupathi Gudi, and Li-yeng Sung,
*An a posteriori error estimator for a quadratic $C^0$-interior penalty method for the biharmonic problem*, IMA J. Numer. Anal.**30**(2010), no. 3, 777–798. MR**2670114**, DOI 10.1093/imanum/drn057 - Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR**2373954**, DOI 10.1007/978-0-387-75934-0 - Eduardo Casas, Mariano Mateos, and Jean-Pierre Raymond,
*Penalization of Dirichlet optimal control problems*, ESAIM Control Optim. Calc. Var.**15**(2009), no. 4, 782–809. MR**2567245**, DOI 10.1051/cocv:2008049 - Eduardo Casas and Jean-Pierre Raymond,
*Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations*, SIAM J. Control Optim.**45**(2006), no. 5, 1586–1611. MR**2272157**, DOI 10.1137/050626600 - 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**, DOI 10.1007/s10589-007-9056-6 - Sudipto Chowdhury, Thirupathi Gudi, and A. K. Nandakumaran,
*A framework for the error analysis of discontinuous finite element methods for elliptic optimal control problems and applications to $C^0$ IP methods*, Numer. Funct. Anal. Optim.**36**(2015), no. 11, 1388–1419. MR**3418817**, DOI 10.1080/01630563.2015.1068182 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR**0520174** - Klaus Deckelnick and Michael Hinze,
*Convergence of a finite element approximation to a state-constrained elliptic control problem*, SIAM J. Numer. Anal.**45**(2007), no. 5, 1937–1953. MR**2346365**, DOI 10.1137/060652361 - Klaus Deckelnick, Andreas Günther, and Michael Hinze,
*Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains*, SIAM J. Control Optim.**48**(2009), no. 4, 2798–2819. MR**2558321**, DOI 10.1137/080735369 - Willy Dörfler,
*A convergent adaptive algorithm for Poisson’s equation*, SIAM J. Numer. Anal.**33**(1996), no. 3, 1106–1124. MR**1393904**, DOI 10.1137/0733054 - 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**, DOI 10.1051/m2an/1979130403131 - Vivette Girault and Pierre-Arnaud Raviart,
*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR**851383**, DOI 10.1007/978-3-642-61623-5 - P. Grisvard,
*Elliptic problems in nonsmooth domains*, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR**775683** - Andreas Günther and Michael Hinze,
*Elliptic control problems with gradient constraints—variational discrete versus piecewise constant controls*, Comput. Optim. Appl.**49**(2011), no. 3, 549–566. MR**2803864**, DOI 10.1007/s10589-009-9308-8 - 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**, DOI 10.1051/m2an/1991250607111 - Max D. Gunzburger, LiSheng Hou, and Thomas P. Svobodny,
*Boundary velocity control of incompressible flow with an application to viscous drag reduction*, SIAM J. Control Optim.**30**(1992), no. 1, 167–181. MR**1145711**, DOI 10.1137/0330011 - Richard S. Falk,
*Approximation of a class of optimal control problems with order of convergence estimates*, J. Math. Anal. Appl.**44**(1973), 28–47. MR**686788**, DOI 10.1016/0022-247X(73)90022-X - 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**, DOI 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**, DOI 10.1051/cocv:2007057 - Kristina Kohls, Arnd Rösch, and Kunibert G. Siebert,
*A posteriori error analysis of optimal control problems with control constraints*, SIAM J. Control Optim.**52**(2014), no. 3, 1832–1861. MR**3212590**, DOI 10.1137/130909251 - Dmitriy Leykekhman and Matthias Heinkenschloss,
*Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems*, SIAM J. Numer. Anal.**50**(2012), no. 4, 2012–2038. MR**3022208**, DOI 10.1137/110826953 - 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**, DOI 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**, DOI 10.1137/S0036142999352187 - S. May, R. Rannacher, and B. Vexler,
*Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems*, SIAM J. Control Optim.**51**(2013), no. 3, 2585–2611. MR**3070527**, DOI 10.1137/080735734 - C. Meyer and A. Rösch,
*Superconvergence properties of optimal control problems*, SIAM J. Control Optim.**43**(2004), no. 3, 970–985. MR**2114385**, DOI 10.1137/S0363012903431608 - G. Of, T. X. Phan, and O. Steinbach,
*An energy space finite element approach for elliptic Dirichlet boundary control problems*, Numer. Math.**129**(2015), no. 4, 723–748. MR**3317816**, DOI 10.1007/s00211-014-0653-x - 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**, DOI 10.1007/s00211-011-0360-9 - F. Tröltzsch,
*Optimale Steuerung Partieller Differentialgleichungen*, I, Vieweg, Cambridge University Press, 2005. - R. Verfürth,
*A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques*, Wiley-Teubner, Chichester, 1995.

## Additional Information

**Sudipto Chowdhury**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
- Email: sudipto10@math.iisc.ernet.in
**Thirupathi Gudi**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
- Email: gudi@math.iisc.ernet.in
**A. K. Nandakumaran**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India
- Email: nands@math.iisc.ernet.in
- Received by editor(s): May 19, 2015
- Received by editor(s) in revised form: August 27, 2015, and October 7, 2015
- Published electronically: June 20, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1103-1126 - MSC (2010): Primary 65N30, 65N15, 65N12, 65K10
- DOI: https://doi.org/10.1090/mcom/3125
- MathSciNet review: 3614013