Finite and Boundary Element Energy Approximations of Dirichlet Control Problems

  • Günther Of
  • Thanh Xuan Phan
  • Olaf Steinbach
Conference paper


We study a Dirichlet boundary control problem of the Poisson equation where the Dirichlet control is considered in the energy space H 1∕2(Γ). Both, finite and boundary element approximations of the minimal solution are presented. We state the unique solvability of both approaches, as well as the stability and error estimates. The numerical example is in good agreement with the theoretical results.


Boundary Element Boundary Element Method Boundary Integral Equation Element Approximation Boundary Integral Operator 
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  1. 1.
    E. Casas, J. P. Raymond: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45 (2006) 1586–1611.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    K. Deckelnick, A. Günther, M. Hinze: Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J. Control Optim. 48 (2009) 2798–2819.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    M. Hinze, R. Pinnau, M Ulbrich, S. Ulbrich: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23, Springer, Heidelberg, 2009.Google Scholar
  4. 4.
    G. C. Hsiao, W. L. Wendland: Boundary Integral Equations, Springer, Heidelberg, 2008.zbMATHCrossRefGoogle Scholar
  5. 5.
    B. N. Khoromskij, G. Schmidt: Boundary integral equations for the biharmonic Dirichlet problem on non-smooth domains. J. Integral Equation Appls. 11 (1999) 217–253.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    K. Kunisch, B. Vexler: Constrainted Dirichlet boundary control in L 2 for a class of evolution equations. SIAM J. Control Optim. 46 (2007) 1726–1753.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    S. May, R. Rannacher, B. Vexler: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. Lehrstuhl für Angewandte Mathematik, Universität Heidelberg, Preprint 05/2008.Google Scholar
  8. 8.
    G. Of, T. X. Phan, O. Steinbach: An energy space finite element approach for elliptic Dirichlet boundary control problems. Berichte aus dem Institut für Numerische Mathematik, Bericht 2009/13, TU Graz, 2009.Google Scholar
  9. 9.
    G. Of, T. X. Phan, O. Steinbach: Boundary element methods for Dirichlet boundary control problems. Math. Methods Appl. Sci., published online, 2010.Google Scholar
  10. 10.
    G. Of, O. Steinbach: A fast multipole boundary element method for a modified hypersingular boundary integral equation. In: Analysis and Simulation of Multifield Problems (W. L. Wendland, M. Efendiev eds.), Lecture Notes in Applied and Computational Mechanics, vol. 12, Springer, Heidelberg, pp. 163–169, 2003.Google Scholar
  11. 11.
    S. Rjasanow, O. Steinbach: The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2007.zbMATHGoogle Scholar
  12. 12.
    O. Steinbach: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York, 2008.zbMATHCrossRefGoogle Scholar
  13. 13.
    B. Vexler: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28 (2007) 957–973.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Computational MathematicsGraz University of TechnologyGrazAustria

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