We investigate finite element approximations of one-dimensional elliptic control problems. For semidiscretizations and full discretizations with piecewise constant controls we derive error estimates in the maximum norm.
Johann Radon Institute for Computational and Applied, Mathematics (RICAM), A-4040 Linz, Austria
Bibliografia
[1] J.P. Aubin, Behaviour of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods, Ann. Scoula Norm. Sup. Pisa 21 (1967), 599-637.
[2] N. Bräutigam, Diskretisierung elliptischer Steuerungsprobleme, Ph.D. Thesis, Jena 2006.
[3] N. Bräutigam, Discretization of Elliptic Control Problems by Finite Elements, Technical Report, Jena 2006.
[4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland 1987.
[5] C. Groß mann and H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner 2005.
[6] M. Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, Computational Optimization and Applications 30 (2005), 45-61.
[7] K. Malanowski, Convergence of Approximations vs. Regularity of Solutions for Convex, Control-Constrained Optimal Control Problems, Appl. Math. Optim. 8 (1981), 69-95.
[8] C. Meyer and A. Rösch, Superconvergence Properties of Optimal Control Problems, SIAM J. Contr. Opt. 43 (2004), 970-985.
[9] J.A. Nitsche, Ein Kriterium für die Quasioptimalität des Ritzschen Verfahrens, Numerische Mathematk 11 (1968), 346-348.
[10] B. Sendov and V.A. Popov, The Averaged Moduli of Smoothness, Wiley-Interscience 1988.
[11] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, Vieweg 2005.