Journal of Scientific Computing

, Volume 56, Issue 1, pp 14–27 | Cite as

Mimetic Discretizations of Elliptic Control Problems



We investigate the performance of the Mimetic Finite Difference (MFD) method for the approximation of a constraint optimal control problem governed by an elliptic operator. Low-order and high-order mimetic discretizations are considered and a priori error estimates are derived, in a suitable discrete norm, for both the control and the state variables. A wide class of numerical experiments performed on a set of examples selected from the literature assesses the robustness of the MFD method and confirms the convergence analysis.


Optimal control problems Mimetic finite difference method Mixed formulation 



We would like to thank the anonymous Referee for his/her valuable suggestions. This work was partially supported by Azione Integrata Italia-Spagna through the project IT097ABB10.


  1. 1.
    Antonietti, P.F., Beirão da Veiga, L., Lovadina, C., Verani, M.: Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems. MOX Rep. 33 (2011). Submitted for publication Google Scholar
  2. 2.
    Antonietti, P.F., Beirão da Veiga, L., Verani, M.: A mimetic discretization of elliptic obstacle problems. Math. Comput. To appear Google Scholar
  3. 3.
    Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23(2), 201–229 (2002) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beirão da Veiga, L.: A mimetic discretization method for linear elasticity. Modél. Math. Anal. Numér. 44(2), 231–250 (2010) MATHCrossRefGoogle Scholar
  5. 5.
    Beirão da Veiga, L., Lipnikov, K.: A mimetic discretization of the Stokes problem with selected edge bubbles. SIAM J. Sci. Comput. 32(2), 875–893 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Beirão da Veiga, L., Manzini, G.: An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems. Int. J. Numer. Methods Eng. 76(11), 1696–1723 (2008) MATHCrossRefGoogle Scholar
  7. 7.
    Beirão da Veiga, L., Gyrya, V., Lipnikov, K., Manzini, G.: Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228(19), 7215–7232 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113(3), 325–356 (2009) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48(4), 1419–1443 (2010) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Berrone, S., Verani, M.: A new marking strategy for the adaptive finite element approximation of optimal control constrained problems. Optim. Methods Softw. 26(4–5), 747–775 (2011) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994) MATHCrossRefGoogle Scholar
  13. 13.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16(2), 275–297 (2006) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. Modél. Math. Anal. Numér. 43(2), 277–295 (2009) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Cangiani, A., Manzini, G., Russo, A.: Convergence analysis of the mimetic finite difference method for elliptic problems. SIAM J. Numer. Anal. 47(4), 2612–2637 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Casas, E., Tröltzsch, F.: Error estimates for linear-quadratic elliptic control problems. In: Analysis and Optimization of Differential Systems (Constanta, 2002), pp. 89–100. Kluwer Academic, Boston (2003) Google Scholar
  19. 19.
    Chen, Y.: Superconvergence of quadratic optimal control problems by triangular mixed finite element methods. Int. J. Numer. Methods Eng. 75(8), 881–898 (2008) MATHCrossRefGoogle Scholar
  20. 20.
    Chen, Y., Huang, Y., Liu, W., Yan, N.: Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J. Sci. Comput. 42(3), 382–403 (2010) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Deng, K., Chen, Y., Lu, Z.: Higher order triangular mixed finite element methods for semilinear quadratic optimal control problem. Numer. Math. Theor. Meth. Appl. 4(2), 180–196 (2011) MathSciNetMATHGoogle Scholar
  22. 22.
    Evans, G., Blackledge, J., Yardley, P.: Numerical Methods for Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, London (2000) MATHCrossRefGoogle Scholar
  23. 23.
    Falk, R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numér. 13(4), 313–328 (1979) MathSciNetMATHGoogle Scholar
  26. 26.
    Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009) MATHCrossRefGoogle Scholar
  28. 28.
    Lie, R., Ma, H., Liu, S., Yan, N., Tang, T.: Theory and algorithm to optimal control problem (2005).
  29. 29.
    Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971). Translated from the French by S.K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 MATHCrossRefGoogle Scholar
  30. 30.
    Lipnikov, K., Morel, J., Shashkov, M.: Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys. 199(2), 589–597 (2004) MATHCrossRefGoogle Scholar
  31. 31.
    Lipnikov, K., Moulton, J., Svyatskiy, D.: A multilevel multiscale mimetic (M3) method for two-phase flows in porous media. J. Comput. Phys. 227, 6727–6753 (2008) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer. Math. 112(1), 115–152 (2009) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Lipnikov, K., Manzini, G., Brezzi, F., Buffa, A.: The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. J. Comput. Phys. 230(2), 305–328 (2011) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Liska, R., Shashkov, M., Ganza, V.: Analysis and optimization of inner products for mimetic finite difference methods on triangular grid. Math. Comput. Simul. 67, 55–66 (2004) MATHCrossRefGoogle Scholar
  35. 35.
    Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43(3), 970–985 (2004) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw. 21(1), 121–134 (2006) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Tröltzsch, F.: Optimal control of partial differential equations. In: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112. Am. Math. Soc., Providence (2010). Translated from the 2005 German original by Jürgen Sprekels Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Paola F. Antonietti
    • 1
  • Nadia Bigoni
    • 1
  • Marco Verani
    • 1
  1. 1.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

Personalised recommendations