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Adaptive Finite Element Method for Dirichlet Boundary Control of Elliptic Partial Differential Equations

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Abstract

In this paper, we consider the Dirichlet boundary control problem of elliptic partial differential equations, and get a coupling system of the state and adjoint state by cancelling the control variable in terms of the control rule, and prove that this coupling system is equivalent to the known Karush–Kuhn–Tucker (KKT) system. For corresponding finite element approximation, we find a measure of the numerical errors by employing harmonic extension, based on this measure, we develop residual-based a posteriori error analytical technique for the Dirichlet boundary control problem. The derived estimators for the coupling system and the KKT system are proved to be reliable and efficient over adaptive mesh. Numerical examples are presented to validate our theory.

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References

  1. Falk, R.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973)

    Article  MathSciNet  Google Scholar 

  2. Geveci, T.: On the approximation of the solution of an optimal control problem governed by an ellptic equation. Rairo Anal. Numér. 13, 313–328 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Casas, E.: Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM Control Optim. Calc. Var. 8, 345–374 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Fursikov, A.V., Gunzburger, M.D., Hou, L.S.: Boundary value problems and optimal boundary control for the Navier–Stokes system: the two-dimensional case. SIAM J. Control Optim. 36, 852–894 (1998)

    Article  MathSciNet  Google Scholar 

  7. Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31, 193–219 (2005)

    Article  MathSciNet  Google Scholar 

  8. Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45, 1586–1611 (2006)

    Article  MathSciNet  Google Scholar 

  9. Vexler, B.: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28, 957–973 (2007)

    Article  MathSciNet  Google Scholar 

  10. Deckelnick, K., Günther, A., Hinze, M.: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48, 2798–2819 (2009)

    Article  MathSciNet  Google Scholar 

  11. May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51, 2585–2611 (2013)

    Article  MathSciNet  Google Scholar 

  12. Apel, T., Mateos, M., Pfefferer, J., Rösch, A.: Error estimates for Dirichlet control problem in polygonal domains, Math. Control Relat. F., 8, E-print

  13. Of, G., Phan, T.X., Steinbach, O.: An energy space finite element approach for elliptic Dirichlet boundary control problems. Numer. Math. 129, 723–748 (2015)

    Article  MathSciNet  Google Scholar 

  14. Gunzburger, M.D., Hou, L.S., Svobodny, T.P.: Analysis and finite element approximation of optimal control problems for the stationary Navier–Stokes equations with Dirichlet controls. Rairo Modél. Math. Anal. Numér. 25, 711–748 (1991)

    Article  MathSciNet  Google Scholar 

  15. Chowdhury, S., Gudi, T., Nandakumaran, A.K.: Error bounds for a Dirichlet boundary control problem based on energy spaces. Math. Comput. 86, 1103–1126 (2017)

    Article  MathSciNet  Google Scholar 

  16. Gunzburger, M.D., Hou, L.S., 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  Google Scholar 

  17. Gong, W., Yan, N.N.: Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs. SIAM J. Control Optim. 49, 984–1014 (2011)

    Article  MathSciNet  Google Scholar 

  18. Hu, W.W., Shen, J.G., Singler, J.R., Zhang, Y.W., Zheng, X.: A superconvergence hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs. Numer. Math. 144, 375–411 (2020)

    Article  MathSciNet  Google Scholar 

  19. Du, S., Cai, Z.: A finite element method for Dirichlet boundary control of elliptic partial differential equations

  20. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  Google Scholar 

  21. Zhang, Z.: Recovery techniques in finite element methods. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and Algorithms, Mathematics Monographs Series 6, pp. 333–412. Science Publisher, New York (2007)

    Google Scholar 

  22. Verfürth, R.: Robust a posteriori error estimates for stationary convection–diffusion equations. SIAM J. Numer. Anal. 43, 1766–1782 (2005)

    Article  MathSciNet  Google Scholar 

  23. Ainsworth, M., Allendes, A., Barrenechea, G.R., Rankin, R.: Fully computable a posteriori error bounds for stabilized FEM approximations of convection–reaction–diffusion problems in three dimentions. Int. J. Numer. Meth. Fluids 73, 765–790 (2013)

    MATH  Google Scholar 

  24. Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47, 2132–2156 (2009)

    Article  MathSciNet  Google Scholar 

  25. Demlow, A., Hirani, A.N.: A posteriori error estimates for finite element exterior calculus: the deRham complex. Found. Comput. Math. 14, 1337–1371 (2014)

    Article  MathSciNet  Google Scholar 

  26. Du, S., Zhang, Z.: A robust residual-type a posteriori error estimator for convection–diffusion equations. J. Sci. Comput. 65, 138–170 (2015)

    Article  MathSciNet  Google Scholar 

  27. Du, S., Sun, S., Xie, X.: Residual-based a posteriori error estimation for multipoint flux mixed finite element methods. Numer. Math. 134, 197–222 (2016)

    Article  MathSciNet  Google Scholar 

  28. Verfürth, R.: A Review of a Posteriori Error Estimates and Adaptive Mesh Refinement Techniques. Wiley-Teubner, New York (1996)

    MATH  Google Scholar 

  29. 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 

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

    Article  MathSciNet  Google Scholar 

  31. 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 

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

    Article  MathSciNet  Google Scholar 

  33. Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)

    Google Scholar 

  34. 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  Google Scholar 

  35. Kohls, K., Rösch, A., Siebert, K.G.: A posteriori error estimators for control constrained optimal control problems. In: Series, Leugering International, of Numerical Mathematics, vol. 160, et al. (eds.) constrained optimization and optimal control for partial differential equations, pp. 431–443. Birkhäuser/Springer, Basel AG, Basel (2012)

  36. Schneider, R., Wachsmuth, G.: A posteriori error estimation for control-constrained, linear-quadratic optimal control problems. SIAM J. Numer. Anal. 54, 1169–1192 (2016)

    Article  MathSciNet  Google Scholar 

  37. Gong, W., Liu, W.B., Tan, Z.Y., Yan, N.N.: A convergent adaptive finite element method for elliptic Dirichlet boundary control problems. IMA J. Numer. Anal. 39, 1985–2015 (2019)

    Article  MathSciNet  Google Scholar 

  38. Bartels, S., Carstensen, C., Dolzmann, G.: Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99, 1–24 (2004)

    Article  MathSciNet  Google Scholar 

  39. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  40. Clemént, P.H.: Approximation by finite element functions using local regularization. Rairosér. Rouge Anal. Numér. 2, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  41. Sangalli, G.: Robust a-posteriori estimator for advection–diffusion–reaction problems. Math. Comput. 77, 41–70 (2008)

    Article  MathSciNet  Google Scholar 

  42. Verfürth, R.: A posteriori error estimates and adaptive mesh-refinment techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    Article  MathSciNet  Google Scholar 

  43. Verfürth, R.: A posteriori error estimates for nonlinear problems: finite element discretizations of elliptic equations. Math. Comput. 62, 445–475 (1994)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Shaohong Du

  2. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Zhiqiang Cai

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  1. Shaohong Du
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  2. Zhiqiang Cai
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Correspondence to Shaohong Du.

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This work is supported in part by the Natural Science Foundation of Chongqing (cstc2018jcyjAX490), the Education Science Foundation of Chongqing (KJZD-K201900701), and the Team Building Projection for Graduate Tutors in Chongqing (JDDSTD201802).

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Du, S., Cai, Z. Adaptive Finite Element Method for Dirichlet Boundary Control of Elliptic Partial Differential Equations. J Sci Comput 89, 36 (2021). https://doi.org/10.1007/s10915-021-01644-3

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  • DOI: https://doi.org/10.1007/s10915-021-01644-3

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