Abstract
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.
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References
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.
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.
M. Hinze, R. Pinnau, M Ulbrich, S. Ulbrich: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23, Springer, Heidelberg, 2009.
G. C. Hsiao, W. L. Wendland: Boundary Integral Equations, Springer, Heidelberg, 2008.
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.
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.
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.
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.
G. Of, T. X. Phan, O. Steinbach: Boundary element methods for Dirichlet boundary control problems. Math. Methods Appl. Sci., published online, 2010.
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.
S. Rjasanow, O. Steinbach: The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2007.
O. Steinbach: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York, 2008.
B. Vexler: Finite element approximation of elliptic Dirichlet optimal control problems. Numer. Funct. Anal. Optim. 28 (2007) 957–973.
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Of, G., Phan, T.X., Steinbach, O. (2012). Finite and Boundary Element Energy Approximations of Dirichlet Control Problems. In: Bock, H., Hoang, X., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25707-0_18
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DOI: https://doi.org/10.1007/978-3-642-25707-0_18
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Online ISBN: 978-3-642-25707-0
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