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

  • Qian Zhang
  • Tengjin Zhao
  • Zhiyue ZhangEmail author
Original Paper
  • 3 Downloads

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.

Keywords

PDE-constrained optimization Interface problem Unfitted mesh 

Notes

Acknowledgements

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

Funding information

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).

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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn. Springer, Berlin (2008)CrossRefGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casas, E., Kunisch, K.: Optimal control of semilinear elliptic equations in measure spaces. SIAM J. Control Optim. 52, 339–364 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chern, I., Shu, Y.: A coupling interface method for elliptic interface problems. J Comput. Phys. 225, 2138–2174 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, vol. 23. Springer, Berlin (2008)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 230–254 (1998)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefGoogle Scholar
  27. 27.
    Liu, X., Sideris, T.: Convergence of the ghost fluid method for elliptic equations with interfaces. Math Comput. 72, 1731–1746 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Massjung, R.: An unfitted discontinuous Galerkin method applied to elliptic interface problems. SIAM J. Numer. Anal. 50, 3134–3162 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Meyer, C., Rösch, A: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ying, W., Wang, W.: A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. 252, 606–624 (2013)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Information TechnologyNanjing University of Chinese MedicineNanjingChina
  2. 2.School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCSNanjing Normal UniversityNanjingChina

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