Two fast finite difference schemes for elliptic Dirichlet boundary control problems

Original Research


In this paper, we propose and analyze two standard finite difference schemes (called Scheme I and Scheme II) for discretizing the first-order necessary optimality systems, which characterize the optimal solutions of Dirichlet boundary control problems governed by elliptic equations. We proved that the proposed schemes are uniformly stable on a uniform mesh, which implies a second-order and first-order convergence of the Scheme I and Scheme II, respectively, provided the optimal solutions have the required regularity. The resulting symmetric indefinite sparse linear systems are solved by the preconditioned GMRES iterative solver with a fast (FFT-based) constraint preconditioner, which numerically shows a mesh-independent convergence rate. Numerical examples, including the case with less regular solutions, are presented to validate our theoretical analysis and demonstrate the promising approximation accuracy and computational efficiency of our proposed schemes and preconditioned iterative solver, respectively. Our developed fast finite difference schemes achieve a comparable order of convergence as the other available schemes in the literature.


Dirichlet boundary control Elliptic PDE Finite difference method Constraint preconditioner 

Mathematics Subject Classification

65N06 65N12 65N22 65F08 65F10 



The author would like to thank the editor and two anonymous referees for their valuable comments and detailed suggestions that have significantly contributed to improving the presentation of this paper.


  1. 1.
    Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  3. 3.
    John, C., Wachsmuth, D.: Optimal Dirichlet boundary control of stationary Navier–Stokes equations with state constraint. Numer. Funct. Anal. Optim. 30(11–12), 1309–1338 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casas, E., Günther, A., Mateos, M.: A paradox in the approximation of Dirichlet control problems in curved domains. SIAM J. Control Optim. 49(5), 1998–2007 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Apel, T., Mateos, M., Pfefferer, J., Rösch, A.: On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim. 53(6), 3620–3641 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Apel, T., Nicaise, S., Pfefferer, J.: Discretization of the poisson equation with non-smooth data and emphasis on non-convex domains. Numer. Methods Partial Differ. Equ. 32(5), 1433–1454 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Apel, T., Mateos, M., Pfefferer, J., Rösch, A.: Error estimates for Dirichlet control problems in polygonal domains: quasi-uniform meshes. Math. Control Relat. Fields 8(1), 217–245 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casas, E., Raymond, J.-P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45(5), 1586–1611 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Vexler, B.: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28(7–8), 957–973 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48(4), 2798–2819 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gong, W., Yan, N.: Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs. SIAM J. Control Optim. 49(3), 984–1014 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51(3), 2585–2611 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Of, G., Phan, T.X., Steinbach, O.: An energy space finite element approach for elliptic Dirichlet boundary control problems. Numerische Mathematik 129(4), 723–748 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gong, W., Liu, W., Tan, Z., Yan, N.: A convergent adaptive finite element method for elliptic Dirichlet boundary control problems. IMA J. Numer. Anal. 22, 1–31 (2018). Google Scholar
  15. 15.
    Mateos, M.: Optimization methods for Dirichlet control problems. Optimization 67(5), 585–617 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Karkulik, M.: A finite element method for elliptic Dirichlet boundary control problems. arXiv e-prints arXiv:1811.09251
  17. 17.
    Of, G., Phan, T.X., Steinbach, O.: Boundary element methods for Dirichlet boundary control problems. Math. Methods Appl. Sci. 33(18), 2187–2205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    John, L., Swierczynski, P., Wohlmuth, B.: Energy corrected FEM for optimal Dirichlet boundary control problems. Numerische Mathematik 139(4), 913–938 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gerdts, M., Greif, G., Pesch, H.J.: Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time. Math. Comput. Simul. 79(4), 1020–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ervedoza, S., Zuazua, E.: The wave equation: control and numerics. In: Cannarsa, P., Coron, J.M. (eds.) Control of Partial Differential Equations, pp. 245–339. Springer, Berlin (2012)CrossRefGoogle Scholar
  22. 22.
    Gugat, M., Sokolowski, J.: A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains. Appl. Anal. 92(10), 2200–2214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Apel, T., Nicaise, S., Pfefferer, J.: Adapted numerical methods for the poisson equation with \({L}^2\) boundary data in NonConvex domains. SIAM J. Numer. Anal. 55(4), 1937–1957 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Heinrich, B.: Finite Difference Methods on Irregular Networks. Birkhuser, Basel (1987)CrossRefzbMATHGoogle Scholar
  25. 25.
    Jovanović, B.S., Süli, E.: Analysis of Finite Difference Schemes. Springer Series in Computational Mathematics, vol. 46. Springer, London (2014)CrossRefGoogle Scholar
  26. 26.
    Chang, L., Gong, W., Yan, N.: Weak boundary penalization for Dirichlet boundary control problems governed by elliptic equations. J. Math. Anal. Appl. 453(1), 529–557 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, New York (2009)zbMATHGoogle Scholar
  29. 29.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. AMS, Providence (2010)zbMATHGoogle Scholar
  30. 30.
    Mateos, M., Neitzel, I.: Dirichlet control of elliptic state constrained problems. Comput. Optim. Appl. 63(3), 825–853 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hu, W., Shen, J., Singler, J.R., Zhang, Y., Zheng, X.: A Superconvergent Hybridizable Discontinuous Galerkin Method for Dirichlet Boundary Control of Elliptic PDEs. arXiv e-prints arXiv:1712.02931
  32. 32.
    LeVeque, R.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia (2007)CrossRefzbMATHGoogle Scholar
  33. 33.
    Coco, A., Russo, G.: Finite-difference ghost-point multigrid methods on cartesian grids for elliptic problems in arbitrary domains. J. Comput. Phys. 241, 464–501 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Coco, A., Russo, G.: Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface. J. Comput. Phys. 361, 299–330 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  36. 36.
    Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment, English Edition. Springer, Berlin (2010)Google Scholar
  37. 37.
    Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9(2), 267–293 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Heidel, G., Wathen, A.: Preconditioning for boundary control problems in incompressible fluid dynamics. Numer. Linear Algebra Appl. 26(1), e2218 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  40. 40.
    Schöberl, J., Zulehner, W.: Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29(3), 752–773 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Li, B., Liu, J., Xiao, M.: A fast and stable preconditioned iterative method for optimal control problem of wave equations. SIAM J. Sci. Comput. 37(6), A2508–A2534 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Brandt, A., Livne, O.: Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics. Revised Edition, Classics in Applied Mathematics. SIAM, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  44. 44.
    Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press Inc., San Diego (2001)zbMATHGoogle Scholar
  45. 45.
    Borzì, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. 51(2), 361–395 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Computer Science and Scientific Computing. Academic Press Inc., Boston (1990)Google Scholar
  47. 47.
    Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969–1972 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algorithm Appl. 19(5), 816–829 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pearson, J.W., Stoll, M., Wathen, A.J.: Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 33(4), 1126–1152 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Pearson, J.W., Wathen, A.: Matching Schur complement approximations for certain saddle-point systems. In: Dick, J., Kuo, F.Y., Wozniakowski, H. (eds.) Contemporary Computational Mathematics: A Celebration of the 80th Birthday of Ian Sloan. Springer, Berlin (2018)Google Scholar
  51. 51.
    Bramble, J.H., Hubbard, B.E.: On the formulation of finite difference analogues of the Dirichlet problem for poisson’s equation. Numer. Math. 4(1), 313–327 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ulbrich, M.: Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces. SIAM, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  54. 54.
    Neittaanmäki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems. Marcel Dekker Inc., New York (1994)zbMATHGoogle Scholar

Copyright information

© Korean Society for Informatics and Computational Applied Mathematics 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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