Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter August 5, 2020

A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

  • Michael Karkulik EMAIL logo

Abstract

We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in H 1 / 2 ( Γ ) . To avoid computing the latter norm numerically, we realize it using the H 1 ( Ω ) norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the H 1 and L 2 norm are proven. We also consider and analyze the case of control constrained problems.

MSC 2010: 65N30

Award Identifier / Grant number: 1170672

Funding statement: Supported by CONICYT through FONDECYT project 1170672.

Acknowledgements

The author would like to thank his colleagues Alejandro Allendes and Enrique Otárola for fruitful discussions.

References

[1] T. Apel, M. Mateos, J. Pfefferer and A. Rösch, On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains, SIAM J. Control Optim. 53 (2015), no. 6, 3620–3641. 10.1137/140994186Search in Google Scholar

[2] T. Apel, M. Mateos, J. Pfefferer and A. Rösch, Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes, Math. Control Relat. Fields 8 (2018), no. 1, 217–245. 10.3934/mcrf.2018010Search in Google Scholar

[3] T. Apel, S. Nicaise and J. Pfefferer, Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains, Numer. Methods Partial Differential Equations 32 (2016), no. 5, 1433–1454. 10.1002/num.22057Search in Google Scholar

[4] F. Ben Belgacem, H. El Fekih and H. Metoui, Singular perturbation for the Dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal. 37 (2003), no. 5, 883–850. 10.1051/m2an:2003057Search in Google Scholar

[5] P. Benner and H. Yücel, Adaptive symmetric interior penalty Galerkin method for boundary control problems, SIAM J. Numer. Anal. 55 (2017), no. 2, 1101–1133. 10.1137/15M1034507Search in Google Scholar

[6] D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed., Cambridge University, Cambridge, 2007. 10.1017/CBO9780511618635Search in Google Scholar

[7] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[8] C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187–1202. 10.1051/m2an:1999140Search in Google Scholar

[9] E. Casas, M. Mateos and J.-P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM Control Optim. Calc. Var. 15 (2009), no. 4, 782–809. 10.1051/cocv:2008049Search in Google Scholar

[10] E. Casas and J.-P. 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. 10.1137/050626600Search in Google Scholar

[11] L. Chang, W. Gong and N. Yan, Weak boundary penalization for Dirichlet boundary control problems governed by elliptic equations, J. Math. Anal. Appl. 453 (2017), no. 1, 529–557. 10.1016/j.jmaa.2017.04.016Search in Google Scholar

[12] S. Chowdhury, T. Gudi and A. K. Nandakumaran, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp. 86 (2017), no. 305, 1103–1126. 10.1090/mcom/3125Search in Google Scholar

[13] J. C. De los Reyes, Numerical PDE-constrained Optimization, Springer Briefs Optim., Springer, Cham, 2015. 10.1007/978-3-319-13395-9Search in Google Scholar

[14] K. Deckelnick, A. Günther and M. 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. 10.1137/080735369Search in Google Scholar

[15] R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), 963–971. 10.1090/S0025-5718-1974-0391502-8Search in Google Scholar

[16] S. Funken, D. Praetorius and P. Wissgott, Efficient implementation of adaptive P1-FEM in Matlab, Comput. Methods Appl. Math. 11 (2011), no. 4, 460–490. 10.2478/cmam-2011-0026Search in Google Scholar

[17] A. V. Fursikov, M. D. Gunzburger and L. S. Hou, Boundary value problems and optimal boundary control for the Navier–Stokes system: The two-dimensional case, SIAM J. Control Optim. 36 (1998), no. 3, 852–894. 10.1137/S0363012994273374Search in Google Scholar

[18] T. 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. 10.1051/m2an/1979130403131Search in Google Scholar

[19] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar

[20] R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, Stud. Math. Appl. 8, North-Holland, Amsterdam, 1981. Search in Google Scholar

[21] W. Gong, W. Liu, Z. Tan and N. Yan, A convergent adaptive finite element method for elliptic Dirichlet boundary control problems, IMA J. Numer. Anal. 39 (2019), no. 4, 1985–2015. 10.1093/imanum/dry051Search in Google Scholar

[22] W. Gong and N. Yan, Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs, SIAM J. Control Optim. 49 (2011), no. 3, 984–1014. 10.1137/100795632Search in Google Scholar

[23] M. D. Gunzburger, L. Hou and T. 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. 10.1137/0330011Search in Google Scholar

[24] R. H. W. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal. 31 (1994), no. 2, 301–323. 10.1137/0731016Search in Google Scholar

[25] C. John and D. Wachsmuth, Optimal Dirichlet boundary control of stationary Navier–Stokes equations with state constraint, Numer. Funct. Anal. Optim. 30 (2009), no. 11–12, 1309–1338. 10.1080/01630560903499001Search in Google Scholar

[26] T. Kärkkäinen, K. Kunisch and P. Tarvainen, Augmented Lagrangian active set methods for obstacle problems, J. Optim. Theory Appl. 119 (2003), no. 3, 499–533. 10.1023/B:JOTA.0000006687.57272.b6Search in Google Scholar

[27] K. Kunisch and B. Vexler, Constrained Dirichlet boundary control in L 2 for a class of evolution equations, SIAM J. Control Optim. 46 (2007), no. 5, 1726–1753. 10.1137/060670110Search in Google Scholar

[28] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer, New York, 1971. 10.1007/978-3-642-65024-6Search in Google Scholar

[29] M. Mateos and I. Neitzel, Dirichlet control of elliptic state constrained problems, Comput. Optim. Appl. 63 (2016), no. 3, 825–853. 10.1007/s10589-015-9784-ySearch in Google Scholar

[30] 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. 10.1137/080735734Search in Google Scholar

[31] R. H. Nochetto and L. B. Wahlbin, Positivity preserving finite element approximation, Math. Comp. 71 (2002), no. 240, 1405–1419. 10.1090/S0025-5718-01-01369-2Search in Google Scholar

[32] 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. 10.1007/s00211-014-0653-xSearch in Google Scholar

[33] J. Pfefferer and M. Winkler, Finite element error estimates for normal derivatives on boundary concentrated meshes, SIAM J. Numer. Anal. 57 (2019), no. 5, 2043–2073. 10.1137/18M1181341Search in Google Scholar

[34] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. 10.1090/S0025-5718-1990-1011446-7Search in Google Scholar

[35] F. Tröltzsch, Optimal Control of Partial Differential Equations, Grad. Stud. Math. 112, American Mathematical Society, Providence, 2010. 10.1090/gsm/112Search in Google Scholar

[36] M. Winkler, Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization, Numer. Math. 144 (2020), no. 2, 413–445. 10.1007/s00211-019-01091-1Search in Google Scholar

Received: 2019-07-02
Revised: 2020-06-08
Accepted: 2020-06-17
Published Online: 2020-08-05
Published in Print: 2020-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.2.2024 from https://www.degruyter.com/document/doi/10.1515/cmam-2019-0104/html
Scroll to top button