Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

¹ University of Sussex Department of Mathematics Mantell Building Falmer, Brighton BN1 9RF, UK. michael.hintermueller@uni-graz.at
² Karl-Franzens-University of Graz Department of Mathematics and Scientific Computing Heinrichstrasse 36, 8010 Graz, Austria. ian.kopacka@uni-graz.at; stefan.volkwein@uni-graz.at

Received: 13 December 2006
Revised: 7 November 2007
Revised: 31 March 2008

Abstract

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners.