Home A Finite Element Method for Elliptic Dirichlet Boundary Control Problems
Article
Licensed
Unlicensed Requires Authentication

A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

  • Michael Karkulik EMAIL logo
Published/Copyright: August 5, 2020

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

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Modern Problems of Numerical Analysis. On the Centenary of the Birth of Alexander Andreevich Samarskii
  4. Special Issue Articles
  5. Finite Difference Approximation of a Generalized Time-Fractional Telegraph Equation
  6. Weighted Estimates for Boundary Value Problems with Fractional Derivatives
  7. Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems
  8. Reliable Computer Simulation Methods for Electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation
  9. Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources
  10. On Convergence of Difference Schemes for Dirichlet IBVP for Two-Dimensional Quasilinear Parabolic Equations with Mixed Derivatives and Generalized Solutions
  11. Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation
  12. A Finite Element Splitting Method for a Convection-Diffusion Problem
  13. Incomplete Iterative Implicit Schemes
  14. Explicit Runge–Kutta Methods Combined with Advanced Versions of the Richardson Extrapolation
  15. Regular Research Articles
  16. A General Superapproximation Result
  17. A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem
  18. A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems
  19. A New Mixed Functional-probabilistic Approach for Finite Element Accuracy
  20. Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem
  21. A Finite Element Method for Elliptic Dirichlet Boundary Control Problems
Downloaded on 25.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/cmam-2019-0104/html
Scroll to top button