Achieving optimal convergence order for FEM in control constrained optimal control problems

We propose a new heuristic approach to overcome convergence order restrictions implied by the low regularity of the optimal control due to the activity interface of control constraints. Aligning the mesh with the interface yields an improved approximation of the control. Utility of the approach is demonstrated by numerical experiments. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

[1]  Kunibert G. Siebert,et al.  A Posteriori Error Analysis of Optimal Control Problems with Control Constraints , 2014, SIAM J. Control. Optim..

[2]  Wenbin Liu,et al.  A Posteriori Error Estimates for Distributed Convex Optimal Control Problems , 2001, Adv. Comput. Math..

[3]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for PDE-Constrained Optimal Control Problems , 2007 .

[4]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[5]  Kunibert G. Siebert,et al.  A Posteriori Error Estimators for Control Constrained Optimal Control Problems , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[6]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

[7]  René Schneider,et al.  A Posteriori Error Estimation for Control-Constrained, Linear-Quadratic Optimal Control Problems , 2016, SIAM J. Numer. Anal..

[8]  F. Tröltzsch Optimale Steuerung partieller Differentialgleichungen , 2005 .

[9]  Boris Vexler,et al.  Adaptive Finite Elements for Elliptic Optimization Problems with Control Constraints , 2008, SIAM J. Control. Optim..

[10]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[11]  David Sevilla,et al.  Polynomial integration on regions defined by a triangle and a conic , 2010, ISSAC.

[12]  Michael Hintermüller,et al.  AN A POSTERIORI ERROR ANALYSIS OF ADAPTIVE FINITE ELEMENT METHODS FOR DISTRIBUTED ELLIPTIC CONTROL PROBLEMS WITH CONTROL CONSTRAINTS , 2008 .