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Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

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Abstract

This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L 2(Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.

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References

  1. Apel, Th.: Interpolation of non-smooth functions on anisotropic finite element meshes. Math. Model. Numer. Anal. 33, 1149–1185 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apel, Th., Sirch, D.: L 2-error estimates for the Dirichlet- and Neumann problem on anisotropic finite element meshes. Preprint SPP1253-02-06, DFG Priority Program 1253, Erlangen (2008). Appl. Math. (accepted for publication)

  3. Apel, Th., Winkler, G.: Optimal control under reduced regularity. Appl. Numer. Math. (2008). doi:10.1016/j.apnum.2008.12.003

    Google Scholar 

  4. Apel, Th., Nicaise, S., Schöberl, J.: Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math. 89, 193–223 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Apel, Th., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Numer. Anal. 21, 843–856 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Apel, Th., 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 

  7. Apel, Th., Sirch, D., Winkler, G.: Error estimates for control constrained optimal control problems: discretization with anisotropic finite element meshes. Preprint SPP1253-02-06, DFG Priority Program 1253, Erlangen (2008)

  8. Apel, Th., Rösch, A., Sirch, D.: L ∞-error estimates on graded meshes with application to optimal control. SIAM J. Control Optim. 48(3), 1771–1796 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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  MATH  Google Scholar 

  10. Bochev, P., Gunzburger, M.: Least-squares finite-element methods for optimization and control problemes for the Stokes equations. Comput. Math. Appl. 48, 1035–1057 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    MATH  Google Scholar 

  12. Casas, E.: Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math. 26, 137–153 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Casas, E., Tröltzsch, F.: Error estimates for linear-quadratic elliptic control problems. In: Barbu, V., et al. (eds.) Analysis and Optimization of Differential Systems, pp. 89–100. Kluwer Academic, Boston (2003)

    Google Scholar 

  14. 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  MATH  Google Scholar 

  15. Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations. SIAM J. Control Optim. 46, 952–982 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Y.: Superconvergence of mixed finite element methods for optimal control problems. Math. Comput. 77(263), 1269–1291 (2008)

    Article  MATH  Google Scholar 

  17. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Reprinted by SIAM, Philadelphia (2002)

    MATH  Google Scholar 

  18. Dauge, M.: Stationary Stokes and Navier-Stokes systems on two- and three-dimensional domains with corners. Part I: Linearized equations. SIAM J. Math. Anal. 20, 27–52 (1989)

    Article  MathSciNet  Google Scholar 

  19. Deckelnick, K., Hinze, M.: Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations. Numer. Math. 97, 297–320 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Falk, M.: Approximation 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 

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

    MathSciNet  MATH  Google Scholar 

  22. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

    MATH  Google Scholar 

  23. Gunzburger, M., Hou, L., Svobodny, T.: Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. Math. Model. Numer. Anal. 25, 711–748 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Gunzburger, M., Hou, L., Svobodny, T.: Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls. Math. Comput. 57, 123–151 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13, 321–334 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lazaar, J., Nicaise, S.: A non-conforming finite element method with anisotropic mesh grading for the incompressible Navier-Stokes equations in domains with edges. Calcolo 39, 123–168 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Maz’ya, V.G., Plamenevskiĭ, B.A.: The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries, part I, II. Z. Anal. Anwend. 2, 335–359, 523–551 (1983) (In Russian)

    Google Scholar 

  30. Maz’ya, V.G., Rossmann, J.: Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains. Math. Methods Appl. Sci. 29, 965–1017 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Meyer, C., Rösch, A.: L ∞-estimates for approximated optimal control problems. SIAM J. Control Optim. 44, 1636–1649 (2005)

    Article  MathSciNet  Google Scholar 

  33. Nicaise, S.: Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc., S. Stevin 4, 411–429 (1997)

    MathSciNet  MATH  Google Scholar 

  34. Raugel, G.: Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris, Sér. A 286(18), A791–A794 (1978)

    MathSciNet  Google Scholar 

  35. Rösch, A.: Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwend. 23, 353–376 (2004)

    Article  MATH  Google Scholar 

  36. Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw. 21(1), 121–134 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rösch, A., Vexler, B.: Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing. SIAM J. Numer. Anal. 44(5), 1903–1920 (2006)

    Article  Google Scholar 

  38. Roßmann, J.: Gewichtete Sobolev–Slobodetskiĭ–Räume und Anwendungen auf elliptische Randwertaufgaben in Gebieten mit Kanten. Habilitationsschrift, Universität Rostock (1988)

  39. Winkler, G.: Control constrained optimal control problems in non-convex three dimensional polyhedral domains. PhD thesis, TU Chemnitz (2008). http://archiv.tu-chemnitz.de/pub/2008/0062

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Authors and Affiliations

  1. LAMAV, Institut des Sciences et Techniques de Valenciennes, Université de Valenciennes et du Hainaut Cambrésis, B.P. 311, 59313, Valenciennes Cedex, France

    Serge Nicaise

  2. Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85579, Neubiberg, Germany

    Dieter Sirch

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  1. Serge Nicaise
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Correspondence to Serge Nicaise.

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Nicaise, S., Sirch, D. Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity. Comput Optim Appl 49, 567–600 (2011). https://doi.org/10.1007/s10589-009-9305-y

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