A Neumann boundary control problem for a second order elliptic state equation is considered. The problem is regularized by an energy term which is equivalent to the H–1/2(Γ) -norm of the control. Both the unconstrained and the control constrained cases are investigated. The regularity of the state, control, and co-state variables is studied with particular focus on the singularities due to the corners of the two-dimensional domain.
The state and co-state are approximated by piecewise linear finite elements. For the approximation of the control variable we use carefully designed spaces of piecewise linear or piecewise constant functions, such that an inf-sup condition is satisfied. Error estimates for the approximate solution are proved for all three variables and we show a relation between convergence rate and the opening angles at corners of the domain. As the control grows in general unboundedly near the concave corners for unconstrained problems, it becomes active and hence regular when control constraints are present. We show that in this case the convergence rates are higher than in the unbounded case. Numerical tests suggest that the estimates derived are optimal in the unconstrained case but too pessimistic in the control constrained case.
We acknowledge the financial support of the German Research Foundation (DFG) within the International Research Training Group (IGDK) 1754. We further thank Johannes Pfefferer and Arnd Rösch for the fruitful discussions.
 T. Apel, M. Mateos, J. Pfefferer, and A. Rösch. On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim., 53(6):3620–3641, 2015.10.1137/140994186Search in Google Scholar
 T. Apel, J. Pfefferer, and A. Rösch. Finite element error estimates on the boundary with application to optimal control. Math. Comput., 84(291):33–70, 2015.10.1090/S0025-5718-2014-02862-7Search in Google Scholar
 E. Casas, M. Mateos, and F. Tröltzsch. Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems. Comp. Opt. and Appl., 31(2):193–220, 2005.10.1007/s10589-005-2180-2Search in Google Scholar
 P. Grisvard. Elliptic problems in nonsmooth domains. Pitman, Boston, 1985.Search in Google Scholar
 P. Grisvard. Singularities in boundary value problems. Paris: Masson; Berlin: Springer-Verlag, 1992.Search in Google Scholar
 M. Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl., 30:45–61, 2005.10.1007/s10589-005-4559-5Search in Google Scholar
 V. A. Kondrat’ev. Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs., 16:209–292, 1967.Search in Google Scholar
 J.L. Lions. Optimal control of systems governed by partial differential equations. Grundlehren der mathematischen Wissenschaften. Springer, New York, 1971.10.1007/978-3-642-65024-6Search in Google Scholar
 M. Moussaoui and K. Khodja. Régularité des solutions d’un problème mêlé Dirichlet–Signorini dans un domaine polygonal plan. Comm. Partial Differential Equations, 17(5-6):805–826, 1992.10.1080/03605309208820864Search in Google Scholar
 M. Mateos and A. Rösch. On saturation effects in the Neumann boundary control of elliptic optimal control problems. Comput. Optim. Appl., 49(2):359–378, 2011.10.1007/s10589-009-9299-5Search in Google Scholar
 S. May, R. Rannacher, and B. Vexler. Error Analysis for a Finite Element Approximation of Elliptic Dirichlet Boundary Control Problems. SIAM J. Control and Optimization, 51(3):2585–2611, 2013.10.1137/080735734Search in Google Scholar
 V.G. Maz’ya and B.A. Plamenevskij. Weighted spaces with nonhomogeneous norms and boundary value problems in domains with conical points. Am. Math. Soc. Transl. Ser. 2, 123:89–107, 1984.10.1090/trans2/123/03Search in Google Scholar
 G. Of, TX. Phan, and O. Steinbach. An energy space finite element approach for elliptic Dirichlet boundary control problems. Numer. Math., 129(4):723–748, 2015.10.1007/s00211-014-0653-xSearch in Google Scholar
 J. Pfefferer. Numerical analysis for elliptic Neumann boundary control problems on polygonal domains. PhD thesis, Universität der Bundeswehr München, 2014.Search in Google Scholar
 B.I. Wohlmuth. Discretization Methods and Iterative Solvers Based on Domain Decomposition., volume 17 of Lecture Notes in Computational Science and Engineering. Springer, 2001.10.1007/978-3-642-56767-4Search in Google Scholar
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