An adaptive gradient-DWR FE algorithm for an optimal control constrained problem

Abstract. We present an adaptive finite element algorithm for the numerical approximation of distributed control constrained problems governed by second order elliptic PDEs. The algorithm is based on a suitable co-operation between a gradient type descent numerical scheme and the dual weighted residual (DWR) method. We assess its efficiency on several test problems and compare its performances with the ones of the well-known residual based adaptive algorithm, see e.g. [?, ?].

[1]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[2]  Alfio Quarteroni,et al.  Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations , 2005 .

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

[4]  Peter Oswald,et al.  Error estimates and adaptivity , 1994 .

[5]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[6]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[7]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[8]  Roland Becker,et al.  Estimating the control error in discretized PDE-constrained optimization , 2006, J. Num. Math..

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

[10]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[11]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[12]  Karl Kunisch,et al.  A Comparison of a Moreau-Yosida-Based Active Set Strategy and Interior Point Methods for Constrained Optimal Control Problems , 2000, SIAM J. Optim..

[13]  Ronald H. W. Hoppe,et al.  Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints , 2007 .

[14]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[15]  K. Kunisch,et al.  Primal-Dual Strategy for Constrained Optimal Control Problems , 1999 .

[16]  Roland Becker,et al.  Optimal control of the convection-diffusion equation using stabilized finite element methods , 2007, Numerische Mathematik.

[17]  Kunibert G. Siebert,et al.  A Unilaterally Constrained Quadratic Minimization with Adaptive Finite Elements , 2007, SIAM J. Optim..

[18]  Stefano Berrone,et al.  An Adaptive WEM Algorithm for Solving Elliptic Boundary Value Problems in Fairly General Domains , 2006, SIAM J. Sci. Comput..

[19]  Michael Hintermüller,et al.  Goal-oriented adaptivity in control constrained optimal control of partial differential equations , 2008, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[20]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[21]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[22]  Ruo Li,et al.  Adaptive Finite Element Approximation for Distributed Elliptic Optimal Control Problems , 2002, SIAM J. Control. Optim..