We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection-diffusion equation, based on a stabilization method applied to the lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting the error on the cost functional into the sum of an iteration error plus a discretization error. Once the former is reduced below a given threshold (and therefore the computed solution is “near” the optimal solution), the adaptive strategy is operated on the discretization error. To prove the effectiveness of the proposed methods, we report some numerical tests, referring to problems in which the control term is the source term of the advection-diffusion equation.
Mots-clés : optimal control problems, partial differential equations, finite element approximation, stabilized lagrangian, numerical adaptivity, advection-diffusion equations
@article{M2AN_2005__39_5_1019_0,
author = {Dede', Luca and Quarteroni, Alfio},
title = {Optimal control and numerical adaptivity for advection-diffusion equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1019--1040},
publisher = {EDP-Sciences},
volume = {39},
number = {5},
year = {2005},
doi = {10.1051/m2an:2005044},
zbl = {1075.49014},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2005044/}
}
TY - JOUR AU - Dede', Luca AU - Quarteroni, Alfio TI - Optimal control and numerical adaptivity for advection-diffusion equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2005 SP - 1019 EP - 1040 VL - 39 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2005044/ DO - 10.1051/m2an:2005044 LA - en ID - M2AN_2005__39_5_1019_0 ER -
%0 Journal Article %A Dede', Luca %A Quarteroni, Alfio %T Optimal control and numerical adaptivity for advection-diffusion equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 1019-1040 %V 39 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2005044/ %R 10.1051/m2an:2005044 %G en %F M2AN_2005__39_5_1019_0
Dede', Luca; Quarteroni, Alfio. Optimal control and numerical adaptivity for advection-diffusion equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 5, pp. 1019-1040. doi : 10.1051/m2an:2005044. http://www.numdam.org/articles/10.1051/m2an:2005044/
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