Skip to main content
SpringerLink
Log in

Local a Posteriori Error Analysis of Finite Element Method for Parabolic Boundary Control Problems

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

Abstract

We derive space-time local a posteriori error estimates of finite element approximations to Neumann boundary control problems governed by parabolic partial differential equations in a convex bounded domain \(\varOmega \subset {\mathbb {R}}^d\; (d\le 3)\) with Lipschitz boundary. We approximate the state and co-state variables by using piecewise linear and continuous finite elements whereas piecewise constant functions are used to approximate the control variable. The time-discretization is based on the backward Euler method. We derive three different reliable local a posteriori error estimates for Neumann boundary control problems with the observations of the boundary state, the distributed state and the final state. Our derived estimators are of local character in the sense that the leading terms of the estimators depend on the small neighbourhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables around the boundary and provide the necessary feedback in terms of the error indicators for the adaptive mesh refinements in the finite element method. Our numerical results validate the effectiveness of the derived estimators.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Becker, R.: Mesh adaptation for stationary flow control. J. Math. Fluid Mech. 3, 317–341 (2001)

    Article  MathSciNet  Google Scholar 

  2. Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39, 113–132 (2000)

    Article  MathSciNet  Google Scholar 

  3. Ben Belgacem, F., Bernardi, C., El Fekih, H.: Dirichlet boundary control for a parabolic equation with a final observation I: A space-time mixed formulation and penalization. Asymptot. Anal. 71, 101–121 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Casas, E., Clason, C., Kunish, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51, 28–63 (2013)

    Article  MathSciNet  Google Scholar 

  5. Ciarlet, P. G., Lions, J. L.: Handbook of Numerical Analysis, Volume II, Finite Element Methods (Part 1), Elsevier Science Publishers B. V., North-Holland (1991)

  6. Gong, W., Hinze, M., Zhou, Z.: A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52, 97–119 (2014)

    Article  MathSciNet  Google Scholar 

  7. Gong, W., Hinze, M., Zhou, Z.: Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic PDEs. J. Sci. Comput. 66, 941–967 (2016)

    Article  MathSciNet  Google Scholar 

  8. Gong, W., Yan, N.: A posteriori error estimates for boundary control problems governed by the parabolic partial differential equations. J. Comput. Math. 27, 68–88 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Hetch, F.: New Development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Hintermüller, M., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM: Control Optim. Calc. Var. 14, 540–560 (2008)

    MATH  Google Scholar 

  11. Hintermüller, M., Hoppe, R.H.W.: Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47, 1721–1743 (2008)

    Article  MathSciNet  Google Scholar 

  12. Hoppe, R.H.W., Iliash, Y., Iyyunni, C., Sweilam, N.H.: A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Math. 14, 57–82 (2006)

    Article  MathSciNet  Google Scholar 

  13. Ito, K., Kunisch, K.: Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces. Nonlinear Anal. 41, 591–616 (2000)

    Article  MathSciNet  Google Scholar 

  14. Kufner, K., John, O., Fucik, S.: Function Spaces. Nordhoff (1977)

  15. Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13, 321–334 (2002)

    Article  MathSciNet  Google Scholar 

  16. Langer, U., Repin, S., Wolfmayr, M.: Functional a posteriori error estimates for time-periodic parabolic optimal control problems. Numer. Funct. Anal. Optim. 37, 1267–1294 (2016)

    Article  MathSciNet  Google Scholar 

  17. Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51, 2797–2821 (2013)

    Article  MathSciNet  Google Scholar 

  18. Lions, J.L.: Contrôle optimal de systemes gouvernés par des équations aux dérivées partielles. Dunod and Gauthier-Villars (1968)

  19. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971)

  20. Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications, vol. 1. Springer (1972)

  21. Liu, W., Yan, N.: A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math. 120, 159–173 (2000)

    Article  MathSciNet  Google Scholar 

  22. Liu, W., Yan, Y.: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39, 73–99 (2001)

    Article  MathSciNet  Google Scholar 

  23. Liu, W., Yan, N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comput. Math. 15, 285–309 (2001)

    Article  MathSciNet  Google Scholar 

  24. Liu, W., Yan, N.: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93, 497–521 (2003)

    Article  MathSciNet  Google Scholar 

  25. Liu, W., Yan, N.: Local a posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 47, 1886–1908 (2009)

    Article  MathSciNet  Google Scholar 

  26. Li, R., Liu, W., Ma, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002)

    Article  MathSciNet  Google Scholar 

  27. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems Part I: Problems without control constraints. SIAM J. Control Optim. 47, 1150–1177 (2008)

    Article  MathSciNet  Google Scholar 

  28. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems Part II: Problems with control constraints. SIAM J. Control Optim. 47, 1301–1329 (2008)

    Article  MathSciNet  Google Scholar 

  29. Rösch, A.: Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwendungen 23, 353–376 (2004)

    Article  MathSciNet  Google Scholar 

  30. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)

    Article  MathSciNet  Google Scholar 

  31. Sun, T., Ge, L., Liu, W.: Equivalent a posteriori error estimates for a constrained optimal control problem governed by parabolic equations. Inter. J. Numer. Anal. Model. 10, 1–23 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Tang, Y., Chen, Y.: Recovery type a posteriori error estimates of fully discrete finite element methods for general convex parabolic optimal control problems, Numer. Math. Theor. Math. Appl. 5, 573–591 (2012)

    MATH  Google Scholar 

  33. Tang, Y., Hua, Y.: Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems. J. Applied Anal. Comput. 4, 295–306 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Tröltzsch, F.: Optimal Control of Partial Differential Equations Theory, Methods and Applications, , vol. 112. American Mathematical Soc (2010)

  35. Xiong, C., Li, Y.: A posteriori error estimates for optimal distributed control governed by the evaluation equations. Appl. Numer. Math. 61, 181–200 (2011)

    Article  MathSciNet  Google Scholar 

  36. Xu, J., Zhou, A.: Local and parallel finite element algorithm based on two-grid discretizations. Math. Comput. 69, 881–909 (2000)

    Article  MathSciNet  Google Scholar 

  37. Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes. Appl. Numer. Math. 61, 181–200 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous reviewer for the valuable suggestions which greatly improve the content of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Ram Manohar & Rajen Kumar Sinha

Authors
  1. Ram Manohar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajen Kumar Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Manohar, R., Sinha, R.K. Local a Posteriori Error Analysis of Finite Element Method for Parabolic Boundary Control Problems. J Sci Comput 91, 17 (2022). https://doi.org/10.1007/s10915-022-01788-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01788-w

Keywords

Mathematics Subject Classification

Search

Navigation