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Error estimates for mixed and hybrid FEM for elliptic optimal control problems with penalizations

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Abstract

Mixed and hybrid finite element discretizations for distributed optimal control problems governed by an elliptic equation are analyzed. A cost functional keeping track of both the state and its gradient is studied. A priori error estimates and super-convergence properties for the continuous and discrete optimal states, adjoint states, and controls will be given. The approximating finite-dimensional systems will be solved by adding penalization terms for the state and the associated Lagrange multipliers. In general, performing optimization, discretization, hybridization, and penalization in any order lead to the same optimality system. Numerical examples based on the Raviart–Thomas finite elements will be presented.

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References

  1. Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, post-processing and error estimates. ESAIM: Math. Model. Numer. Anal. 19, 7–32 (1985). https://doi.org/10.1051/m2an/1985190100071

    Article  MATH  Google Scholar 

  2. Azmi, B., Kunisch, K.: Analysis and performance of the Barzilai–Borwein step-size rules for optimization problems in Hilbert spaces. J. Optimiz. Theory App. 185, 819–844 (2020). https://doi.org/10.1007/s10957-020-01677-y

    Article  MATH  Google Scholar 

  3. Bahriawati, C., Carstensen, C.: Three matlab implementations of the lowest-order Raviart–Thomas Mfem with a posteriori error control. Comput. Methods Appl. Math. 5, 333–361 (2005). https://doi.org/10.2478/cmam-2005-0016

    Article  MathSciNet  MATH  Google Scholar 

  4. Barzilai, J., Borwein, J.M: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988). https://doi.org/10.1093/imanum/8.1.141

    Article  MathSciNet  MATH  Google Scholar 

  5. Bécache, E., Joly, P., Tsogka, C.: An analysis of new mixed finite elements for the approximation of wave propagation problems. SIAM J. Numer. Anal. 37, 1053–1084 (2000). https://doi.org/10.1137/S0036142998345499

    Article  MathSciNet  MATH  Google Scholar 

  6. Bercovier, M.: Perturbation of mixed variational problems. Application to mixed finite element methods. RAIRO Anal. Numér. 12, 211–236 (1978). https://doi.org/10.1051/m2an/1978120302111

    Article  MathSciNet  MATH  Google Scholar 

  7. Brandts, J.H.: Superconvergencea posteriori error estimation for triangular mixed finite elements. Numer. Math. 68, 311–324 (1994). https://doi.org/10.1007/s002110050064

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). https://doi.org/10.1007/978-1-4612-3172-1

    Book  MATH  Google Scholar 

  9. Chen, Y.: Superconvergence of mixed finite element methods for optimal control problems. Math. Comput. 77, 1269–1291 (2008). https://doi.org/10.1090/S0025-5718-08-02104-2

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, Y., Lu, Z.: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 46, 957–965 (2010). https://doi.org/10.1016/j.finel.2010.06.011

    Article  MathSciNet  Google Scholar 

  11. Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: M2AN 50, 635–650 (2016). https://doi.org/10.1051/m2an/2015051

    Article  MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Gopalakhrishnan, J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42, 283–301 (2004). https://doi.org/10.1137/S0036142902417893

    Article  MathSciNet  MATH  Google Scholar 

  13. Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element methods for the wave equation. Comput. Methods Appl. Mech. Eng. 82, 205–222 (1990). https://doi.org/10.1016/0045-7825(90)90165-I

    Article  MathSciNet  MATH  Google Scholar 

  14. Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions. SIAM J. Numer. Anal. 33, 492–504 (1996). https://doi.org/10.1137/0733026

    Article  MathSciNet  MATH  Google Scholar 

  15. Egger, H., Radu, B.: Super-convergence and post-processing for mixed finite element approximations of the wave equation. Numer. Math. 140, 427–447 (2018). https://doi.org/10.1007/s00211-018-0966-2

    Article  MathSciNet  MATH  Google Scholar 

  16. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). https://doi.org/10.1007/978-1-4757-4355-5

    Book  MATH  Google Scholar 

  17. Hou, T.: Error estimates superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations. Numer. Anal. Error Appl. 5, 348–362 (2012). https://doi.org/10.1134/S1995423912040076

    Article  MATH  Google Scholar 

  18. Hu, W., Shen, J., Singler, J.R., Zhang, Y., Zheng, X.: A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs. Numer. Math. 144, 375–411 (2020). https://doi.org/10.1007/s00211-019-01090-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Jenkins, E.W., Riviaère, B., Wheeler, M.F.: A priori error estimates for mixed finite element approximations of the acoustic wave equation. SIAM J. Numer. Anal. 40, 1698–1715 (2002). https://doi.org/10.1137/S0036142901388068

    Article  MathSciNet  MATH  Google Scholar 

  20. Leng, H., Chen, Y.: Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems. Adv. Comput. Math. 47, 30 (2021). https://doi.org/10.1007/s10444-021-09864-9

    Article  MathSciNet  MATH  Google Scholar 

  21. 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). https://doi.org/10.1137/070694016

    Article  MathSciNet  MATH  Google Scholar 

  22. Quarteroni, A., Valli, A.: Numerical Approximations of Partial Differential Equations. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85268-1

    MATH  Google Scholar 

  23. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods, vol. 606 of Lecture Notes in Mathematics. https://doi.org/10.1007/BFb0064470. Springer, Berlin, Heidelberg (1977)

  24. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. American Mathematical Society, Providence, Rhode Island (2010). https://doi.org/10.1090/gsm/112

    MATH  Google Scholar 

  25. Wohlmuth, B.I., Hoppe, R.H.W.: A comparison of a priori error estimators for mixed finite element discretizations by Raviart–Thomas elements. Math. Comp. 68, 1347–1378 (1999). https://doi.org/10.1090/S0025-5718-99-01125-4

    Article  MathSciNet  MATH  Google Scholar 

  26. Xin, X., Chen, Y., Yi, N.: Error estimates of mixed finite element methods for quadratic control problems. J. Comput. Appl. Math. 233, 1812–1820 (2010). https://doi.org/10.1016/j.cam.2009.09.018

    Article  MathSciNet  MATH  Google Scholar 

  27. Xing, X., Chen, Y.: Error estimates of mixed methods for optimal control problems governed by parabolic equations. Int. J. Numer. Method. Eng. 75, 735–754 (2008). https://doi.org/10.1002/nme.2289

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to the referees for their suggestions and comments that led to the improvement of the manuscript.

Funding

This work was partially supported by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program.

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  1. Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600, Baguio, Philippines

    Gilbert Peralta

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  1. Gilbert Peralta
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Peralta, G. Error estimates for mixed and hybrid FEM for elliptic optimal control problems with penalizations. Adv Comput Math 48, 70 (2022). https://doi.org/10.1007/s10444-022-09980-0

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