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Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization

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Abstract

This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy regularization. The regularity of the solution is carefully carved out exploiting weighted Sobolev and Hölder spaces. This allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains. Numerical experiments confirm that the derived convergence rates are sharp.

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References

  1. Agoshkov, V.I., Lebedev, V.I.: Poincaré–Steklov operators and methods of partition of the domain in variational problems. Comput. Process. Syst. 2, 173–227 (1985)

    MATH  Google Scholar 

  2. Apel, Th, Mateos, M., Pfefferer, J., Rösch, A.: On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim. 53(6), 3620–3641 (2015)

    Article  MathSciNet  Google Scholar 

  3. Apel, Th, Mateos, M., Pfefferer, J., Rösch, A.: Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Math. Control Relat. Fields 8(1), 217–245 (2018)

    Article  MathSciNet  Google Scholar 

  4. Apel, Th, Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52(1), 3–28 (2012)

    Article  MathSciNet  Google Scholar 

  5. Apel, Th, Pfefferer, J., Winkler, M.: Error estimates for the postprocessing approach applied to neumann boundary control problems in polyhedral domains. IMA J. Numer. Anal. 38(4), 1984–2025 (2018)

    Article  MathSciNet  Google Scholar 

  6. Apel, Th, Steinbach, O., Winkler, M.: Error estimates for Neumann boundary control problems with energy regularization. J. Numer. Math. 24(4), 207–233 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Casas, E., Raymond, J.-P.: Error estimates for the numerical approximation of Dirichlet boundary control of semilinear elliptic equations. SIAM J. Control Optim. 45(5), 1586–1611 (2006)

    Article  MathSciNet  Google Scholar 

  9. Chowdhury, S., Gudi, T., Nandakumaran, A.K.: Error bounds for a Dirichlet boundary control problem based on energy spaces. Math. Comp. 86(305), 1103–1126 (2017)

    Article  MathSciNet  Google Scholar 

  10. Ciarlet, P.G.: Basic error estimates for elliptic problems. Finite Element Methods. Handbook of Numerical Analysis, vol. 2, pp. 17–352. Elsevier, North-Holland (1991)

    Google Scholar 

  11. Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on sharply varying grids. Math. Comp. 80(273), 1–9 (2011)

    Article  MathSciNet  Google Scholar 

  12. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Totaro, N.S., Waldschmidt, A.V. (eds.) Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1998)

    Google Scholar 

  13. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  14. Gunzburger, M. D., Hou, L. S., Svobodny, Th. P.: Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. ESAIM: Math. Model. Numer. Anal., 25(6):711–748, 1991

  15. Horger, T., Melenk, M., Wohlmuth, B.I.: On optimal L2- and surface flux convergence in FEM. Comput. Vis. Sci. 16(5), 231–246 (2015)

    Article  Google Scholar 

  16. John, L. J.: Optimal boundary control in energy spaces preconditioning and applications. PhD thesis, TU Graz, 2014

  17. John, L.J., Swierczynski, P., Wohlmuth, B.I.: Energy corrected FEM for optimal Dirichlet boundary control problems. Numer. Math. 139(4), 913–938 (2018)

    Article  MathSciNet  Google Scholar 

  18. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  19. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer, New York (1971)

    Book  Google Scholar 

  20. Mateos, M., Neitzel, I.: Dirichlet control of elliptic state constrained problems. Comput. Optim. Appl. 63(3), 825–853 (2016)

    Article  MathSciNet  Google Scholar 

  21. May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51(3), 2585–2611 (2013)

    Article  MathSciNet  Google Scholar 

  22. Maz’ya, V.G., Rossmann, J.: Elliptic Equations in Polyhedral Domains. AMS, Providence (2010)

    Book  Google Scholar 

  23. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  24. Melenk, M., Wohlmuth, B.I.: Quasi-optimal approximation of surface based lagrange multipliers in finite element methods. SIAM J. Numer. Anal. 50(4), 2064–2087 (2012)

    Article  MathSciNet  Google Scholar 

  25. Mikhailov, S.E.: Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients. J. Math. Anal. Appl. 378, 324–342 (2012)

    Article  Google Scholar 

  26. Nazarov, S.A., Plamenevskij, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter, Berlin (1994)

    Book  Google Scholar 

  27. Of, G., Phan, T.X., Steinbach, O.: An energy space finite element approach for elliptic Dirichlet boundary control problems. Numer. Math. 129(4), 723–748 (2015)

    Article  MathSciNet  Google Scholar 

  28. Pfefferer, J.: Numerical analysis for elliptic Neumann boundary control problems on polygonal domains. PhD thesis, Universität der Bundeswehr München, 2014

  29. Pfefferer, J., Winkler, M.: Finite element error estimates for normal derivatives on boundary concentrated meshes. SIAM J. Numer. Anal. 57(5), 2043–2073 (2019)

    Article  MathSciNet  Google Scholar 

  30. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computing. Clarendon Press, Oxford (1999)

    MATH  Google Scholar 

  31. Steinbach, O.: On the stability of the \(L_2\) projection in fractional Sobolev spaces. Numer. Math. 88(2), 367–379 (2001)

    Article  MathSciNet  Google Scholar 

  32. Winkler, M.: Finite element error analysis for neumann boundary control problems on polygonal and polyhedral domains. PhD thesis, Universität der Bundeswehr München, 2015

  33. Wloka, J., Thomas, C.B., Thomas, M.J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  34. Xu, J., Zhang, S.: Preconditioning the Poincaré-Steklov operator by using Green’s function. Math. Comp. 66(217), 125–138 (1997)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author acknowledges the fruitful discussions with Johannes Pfefferer, Hannes Meinlschmidt and Marco Zank during the preparation of the manuscript.

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  1. Faculty of Mathematics, Technische Universität Chemnitz, Chemnitz, Germany

    Max Winkler

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  1. Max Winkler
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Winkler, M. Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization. Numer. Math. 144, 413–445 (2020). https://doi.org/10.1007/s00211-019-01091-1

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  • DOI: https://doi.org/10.1007/s00211-019-01091-1

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