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Optimal control and numerical adaptivityfor advection–diffusionequations

Published online by Cambridge University Press:  15 September 2005

Luca Dede'  and
Alfio Quarteroni
Luca Dede'
Affiliation:
MOX–Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy. [email protected]
Alfio Quarteroni
Affiliation:
MOX–Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy. [email protected] École Polytechnique Fédérale de Lausanne (EPFL), FSB, Chaire de Modelisation et Calcul Scientifique (CMCS), Station 8, 1015, Lausanne, Switzerland. [email protected]
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Abstract

We propose a general approach for the numerical approximation ofoptimal control problems governed by a linear advection–diffusionequation, based on a stabilization method applied to theLagrangian functional, rather than stabilizing the state andadjoint equations separately. This approach yields a coherentlystabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order termsare needless. Our a posteriori estimates stems from splitting theerror on the cost functional into the sum of an iteration errorplus a discretization error. Once the former is reduced below agiven threshold (and therefore the computed solution is “near”the optimal solution), the adaptive strategy is operated on thediscretization error. To prove the effectiveness of the proposedmethods, we report some numerical tests, referring to problems inwhich the control term is the source term of theadvection–diffusion equation.

Keywords

Optimal control problemspartial differentialequationsfinite element approximationstabilizedLagrangiannumerical adaptivityadvection–diffusion equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 1019 - 1040
Copyright
© EDP Sciences, SMAI, 2005

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