Optimal control and numerical adaptivity for advection-diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 5, p. 1019-1040

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.

DOI : https://doi.org/10.1051/m2an:2005044
Classification:  35J25,  49J20,  65N30,  76R50
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {5},
     year = {2005},
     pages = {1019-1040},
     doi = {10.1051/m2an:2005044},
     zbl = {1075.49014},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_5_1019_0}
}
Dede', Luca; Quarteroni, Alfio. Optimal control and numerical adaptivity for advection-diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 5, pp. 1019-1040. doi : 10.1051/m2an:2005044. http://www.numdam.org/item/M2AN_2005__39_5_1019_0/

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