Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance

Abstract

This paper presents a numerical method for the optimal control problem governed by the heat diffusion equation inside a composite medium. The contact resistance at the interface of constitute materials allows for jumps of the temperature field. The derivation process of the Karush-Kuhn-Tucher system is given by the formal Lagrange method. Due to the discontinuity of the temperature field, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore, the unfitted finite element method is applied to discrete the state equation required in the variational discretization approach. Optimal error estimates in the broken H1-norm and L2-norm for the control, state, and adjoint state are derived. Some numerical examples are provided to confirm the theoretical results.

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References

  1. 1.

    An, N., Chen, H.: A partially penalty immersed interface finite element method for anisotropic elliptic interface problems. Numer. Methods Partial Differential Equations 30, 1984–2028 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Apel, T., Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52, 3–28 (2012)

  3. 3.

    Apel, T., Sirch, D.: A Priori Mesh Grading for Distributed Optimal Control Problems Constrained Optimization and Optimal Control for Partial Differential Equations, pp 377–389. Springer, Berlin (2012)

    Google Scholar 

  4. 4.

    Bedrossian, J., Brecht, J., Zhu, S., Sifakis, E., Teran, J.: A second order virtual node method for elliptic problems with interface and irregular domains. J. Comput. Phys. 229, 6405–6426 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Belgacem, F.B., Bernardi, C., Jelassi, F., Brahim, M.M.: Finite element methods for the temperature in composite media with contact resistance. J. Sci Comput. 63(2), 478–501 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn. Springer, Berlin (2008)

    Google Scholar 

  7. 7.

    Butt, M.M., Yuan, Y.: A full multigrid method for distributed control problems constrained by stokes equations. Numer. Math. Theor. Meth Appl. 10, 639–655 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Casas, E., Kunisch, K.: Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52, 339–364 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Chern, I., Shu, Y.: A coupling interface method for elliptic interface problems. J Comput. Phys. 225, 2138–2174 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Fries, T., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Meth. Engng. 84, 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Guan, H.B., Shi, D.Y.: A high accuracy NFEM for constrained optimal control problems governed by elliptic equations. Appl. Math Comput. 245, 382–390 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Hansbo, A., Hansbo, P.: An unfitted finite element based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg. 191, 5537–5552 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    He, X., Lin, T., Lin, Y.: The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer. Methods Partial Differential Equations 28, 312–330 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Hellrung, J., Wang, L., Sifakis, E., Teran, J.: A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J. Comput. Phys. 231, 2015–2048 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

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

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, vol. 23. Springer, Berlin (2008)

    Google Scholar 

  17. 17.

    Hou, S., Wang, W., Wang, L.: Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces. J. Comput. Phys. 229, 7162–7179 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Hou, T., Liu, C., Chen, H.: Fully discrete H1-Galerkin mixed finite element methods for parabolic optimal control problems. Numer. Math. Theor. Meth Appl. 12, 134–153 (2019)

    MATH  Article  Google Scholar 

  19. 19.

    Ji, H., Chen, J., Li, Z.: A high-order source removal finite element method for a class of elliptic interface problems. Appl. Numer Math. 130, 112–130 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Ji, H., Wang, F., Chen, J.: Unfitted finite element methods for the heat conduction in composite media with contact resistance. Numer Methods Partial Differential Equations 33(1), 354–380 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Ji, H., Zhang, Q., Wang, Q., Xie, Y.: A partially penalised immersed finite element method for elliptic interface problems with non-homogeneous jump conditions. East. Asia. J Appl.Math. 8, 1–23 (2018)

    MathSciNet  Article  Google Scholar 

  22. 22.

    LeVeque, R., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 230–254 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Li, Z., Ito, K.: Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci Comput. 23, 1225–1242 (2001)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer Anal. 53, 1121–1144 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Liu, C., Hou, T., Yang, Y.: Superconvergence of H1-Galerkin mixed finite element methods for elliptic optimal control problems. East. Asia. J. Appl Math. 9, 87–101 (2019)

    Article  Google Scholar 

  27. 27.

    Liu, X., Sideris, T.: Convergence of the ghost fluid method for elliptic equations with interfaces. Math Comput. 72, 1731–1746 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Massjung, R.: An unfitted discontinuous Galerkin method applied to elliptic interface problems. SIAM J. Numer. Anal. 50, 3134–3162 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

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

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Negri, F., Rozza, G., Manzoni, A.: Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci Comput. 35, A2316–A2340 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Oevermann, M., Klein, R.: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J. Comput. Phys 219, 749–769 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Ozisik, M.N.: Heat Conduction, 2nd edn. Wiley, New York (1993)

    Google Scholar 

  33. 33.

    Shu, Y., Chern, I., Chang, C.: Accurate gradient approximation for complex interface problems in 3D by an improved coupling interface method. J. Comput. Phys. 275, 642–661 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Ying, W., Henriquez, C.: A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. 227, 1046–1074 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Ying, W., Wang, W.: A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. 252, 606–624 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Yu, S., Zhou, Y., Wei, G.: Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys. 224, 729–756 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Zhang, Q., Ito, K., Li, Z., Zhang, Z.: Immersed finite elements for optimal control problems of elliptic pdes with interfaces. J. Comput. Phys. 298(C), 305–319 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Zhang, Q., Weng, Z., Ji, H., Zhang, B.: Error estimates for an augmented method for one-dimensional elliptic interface problems. Adv. Differ. Equ. 2018, 307 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Zhou, Y., Zhao, S., Feig, M., Wei, G.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213, 1–30 (2006)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

We are very grateful to anonymous referees for their valuable suggestions which have helped to improve the paper.

Funding

This work is partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB110014), the National Natural Science Foundation of China (Grant Nos. 11471166 and 11701291), and the Natural Science Foundation of Jiangsu Province (Grant No. BK20160880).

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Correspondence to Zhiyue Zhang.

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Zhang, Q., Zhao, T. & Zhang, Z. Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance. Numer Algor 84, 165–180 (2020). https://doi.org/10.1007/s11075-019-00750-6

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Keywords

  • PDE-constrained optimization
  • Interface problem
  • Unfitted mesh