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A two-grid discretization scheme for optimal control problems of elliptic equations

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Abstract

In the paper, a two-grid finite element scheme is discussed for distributed optimal control governed by elliptic equations. With this new scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

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References

  1. Apel, T., Rösch, A., Winkler, G.: Optimal control in non-convex domains: a priori discretization error estimates. Calcolo 44, 137–158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic optimal control problem. Comput. Optim. Appl. 23, 201–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axelsson, O., Layton, W.: A two-level discretization of nonlinear boundary value problems. SIAM J. Numer. Anal. 33, 2359–2374 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. Bi, C., Ginting, V.: Two-grid finite volume element method for linear and nonlinear elliptic problems. Numer. Math. 108, 177–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bi, C., Ginting, V.: Two-grid discontinuous Galerkin method for quasi-linear elliptic problems. J. SCi. Comput 49, 311–331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, L., Chen, Y.: Two-grid method for nonlinear reaction-diffusion equation by mixed finite element methods. J. SCi. Comput. 49, 383–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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, 382–403 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciarlet, P.: The finite element methods for elliptic problems. North-Holland, New York (1978)

    MATH  Google Scholar 

  10. Dawson, C.N., Wheeler, M.F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contemp. Math. 180, 191–203 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du, L., Yan, N.: High-accuracy finite element method for optimal control problem. J. Syst. Sci. Complex 14, 106–110 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Falk, F.S.: Approxomation 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 

  15. Girault, V., Raviart, P.: Finite element methods for Navier-stokes equations. Theory and algorithms. Springer, Heidelberg (1996)

    MATH  Google Scholar 

  16. Gunzburger, M.D., Hou, L.: Finite-dimensional approximation of a class of constrained nonlinear optimal control problems. SIAM J. Control Optim. 34, 1001–1043 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gunzburger, M.D., Hou, L., Svobodny, T. P.: Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30, 167–181 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gunzburger, M.D., Hou, L., Svobodny, T.: Optimal control and optimization of viscous, incompressible flows. In: Incompressible computational fluid dynamics, pp 109–150. Cambridge University press, Cambridge (1993)

  19. Haslinger, J., Neittaanmaki, P.: Finite element approximation for optimal shape design. John Wiley, Chichester (1989)

    MATH  Google Scholar 

  20. Hintermüller, M., Hoppe, R.: Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations. SIAM J. Control Optim. 48, 5468–5487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hoppe, R., Kieweg, M.: A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. J. Numer. Math. 17, 219–244 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Hou, L., Turner, J.C.: Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls. Numer. Math. 71, 289–315 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu, X., Cheng, X.: Acceleration of a two-grid method for eigenvalue problems. Math. Comput. 80, 1287–1301 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kohls, K., Rösch, A., Siebert, K.G.: A posteriori error analysis of optimal control problems with control constraints. SIAM J. Control. Optim. 52, 1832–1861 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lasiecka, I.: Ritz-Galerkin approximation of the optimal boundary control problem for parabolic systems with Dirichlet boundary conditions. SIAM J. Control. Optim. 22, 477–500 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  26. Layton, W., Lenferink, W.: Two-level Picard and modified Picard methods for the Navier-Stokes equations. Appl. Math. Comput. 69, 263–274 (1995)

    MathSciNet  MATH  Google Scholar 

  27. Li, Q., Yang, Y.: A two-grid discretization scheme for the Steklov eigenvalue problem. J. Appl. Math. Comput. 36, 129–139 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, R., Liu, W., Yan, N.: A posteriori error estimates of recovery type for distributed convex optimal control problems problems. J. SCi. Comput. 33, 155–182 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lions, J.L.: Optimal control of systems governed by partial differential equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  30. Liu, H., Yan, N.: Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations. J. Comput. Appl. Math. 209, 187–207 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu, W., Yan, N.: A posteriori error estimates for control problems governed by Stokes equations. SIAM J. Numer. Anal. 40, 1850–1869 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, W., Yan, N.: A posteriori error estimates for a model boundary optimal control problem. J. Comput. Appl. Math. 120, 159–173 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, W., Yan, N.: Adaptive finite element methods for optimal control governed by PDEs. Science Press, Beijing (2008)

    Google Scholar 

  35. Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems. Appl. Math. Optim. 8, 69–95 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  36. Marion, M., Xu, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32, 1170–1184 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Naga, A., Zhang, Z., Zhou, A.: Enhancing eigenvalue approximation by gradient recovery. SIAM J. Sci. Comput. 28, 1289–1300 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rösch, A., Vexler, B.: Optimal control of the Stokes equation: a priori error analysis for finite element discretization with postprocessing. SIAM J. Numer. Anal. 44, 1903–1920 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Utnes, T.: Two-grid finite element formulations of the incompressible Navier-Stokes equations. Comm. Numer. Methods Engrg. 34, 675–684 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vexler, B., Wollner, W.: Adaptive finite element for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47, 509–534 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu, L., Allen, M. B.: Two-grid method for mixed finite element solution of coupled reaction-diffusion system. Numer. Methods Partial Differ. Eq 15, 589–604 (1999)

    MATH  Google Scholar 

  43. Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal. 29, 303–319 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu, J.: A novel two-grid method for semilinear equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  45. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer. Anal. 33, 1759–1777 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73, 1139–1152 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70, 17–25 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Xu, J., Zhou, A.: Local and parallel finite element algorithms for eigenvalue problems. Acta Math. Appl. Sin. 18, 185–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  49. Yang, Y., Bi, H.: Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 49, 1602–1624 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China

    Huipo Liu & Shuanghu Wang

Authors
  1. Huipo Liu
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  2. Shuanghu Wang
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Correspondence to Huipo Liu.

Additional information

This work was partially supported by the National Natural Science Foundation of China under Grants 11575028, 91530108, 91330103.

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Liu, H., Wang, S. A two-grid discretization scheme for optimal control problems of elliptic equations. Numer Algor 74, 699–716 (2017). https://doi.org/10.1007/s11075-016-0168-x

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  • DOI: https://doi.org/10.1007/s11075-016-0168-x

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