Skip to main content
Springer Nature Link
Log in

Mimetic Discretizations of Elliptic Control Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  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

  2. Antonietti, P.F., Beirão da Veiga, L., Verani, M.: A mimetic discretization of elliptic obstacle problems. Math. Comput. To appear

  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)

    Article  MathSciNet  Google Scholar 

  4. Beirão da Veiga, L.: A mimetic discretization method for linear elasticity. Modél. Math. Anal. Numér. 44(2), 231–250 (2010)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. Modél. Math. Anal. Numér. 43(2), 277–295 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chen, Y.: Superconvergence of quadratic optimal control problems by triangular mixed finite element methods. Int. J. Numer. Methods Eng. 75(8), 881–898 (2008)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  22. Evans, G., Blackledge, J., Yardley, P.: Numerical Methods for Partial Differential Equations. Springer Undergraduate Mathematics Series. Springer, London (2000)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  26. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, New York (2009)

    Book  MATH  Google Scholar 

  28. Lie, R., Ma, H., Liu, S., Yan, N., Tang, T.: Theory and algorithm to optimal control problem (2005). http://dsec.pku.edu.cn/~rli/research/doc/html/ocp_e.html

  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

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer. Math. 112(1), 115–152 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  35. Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43(3), 970–985 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw. 21(1), 121–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Paola F. Antonietti, Nadia Bigoni & Marco Verani

Authors
  1. Paola F. Antonietti
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Nadia Bigoni
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Marco Verani
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Nadia Bigoni.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Antonietti, P.F., Bigoni, N. & Verani, M. Mimetic Discretizations of Elliptic Control Problems. J Sci Comput 56, 14–27 (2013). https://doi.org/10.1007/s10915-012-9659-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9659-7

Keywords