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Convergence of discontinuous Galerkinapproximations of an optimal control problem associated tosemilinear parabolic PDE's

Published online by Cambridge University Press:  16 December 2009

Konstantinos Chrysafinos
Konstantinos Chrysafinos*
Affiliation:
National Technical University of Athens, Department of Mathematics, Zografou Campus, Athens 15780, Greece. chrysafinos@math.ntua.gr
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Abstract

A discontinuous Galerkin finite element method for an optimal control problem related to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. Convergence of discrete schemes of arbitrary order is proven. In addition, the convergence of discontinuous Galerkin approximations of the associated optimality system to the solutions of the continuous optimality system is shown. The proof is based on stability estimates at arbitrary time points under minimal regularity assumptions, and a discrete compactness argument for discontinuous Galerkin schemes (see Walkington [SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).

Keywords

Discontinuous Galerkin approximationsdistributed controlsstability estimatessemi-linear parabolic PDE's.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 1 , January 2010 , pp. 189 - 206
Copyright
© EDP Sciences, SMAI, 2009

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