# Optimal control and numerical adaptivity for advection–diffusion equations

- Volume: 39, Issue: 5, page 1019-1040
- ISSN: 0764-583X

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topDede', Luca, and Quarteroni, Alfio. "Optimal control and numerical adaptivity for advection–diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 1019-1040. <http://eudml.org/doc/245174>.

@article{Dede2005,

abstract = {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.},

author = {Dede', Luca, Quarteroni, Alfio},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {optimal control problems; partial differential equations; finite element approximation; stabilized lagrangian; numerical adaptivity; advection–diffusion equations; stabilized Lagrangian; advection-diffusion equations.},

language = {eng},

number = {5},

pages = {1019-1040},

publisher = {EDP-Sciences},

title = {Optimal control and numerical adaptivity for advection–diffusion equations},

url = {http://eudml.org/doc/245174},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Dede', Luca

AU - Quarteroni, Alfio

TI - Optimal control and numerical adaptivity for advection–diffusion equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 5

SP - 1019

EP - 1040

AB - 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.

LA - eng

KW - optimal control problems; partial differential equations; finite element approximation; stabilized lagrangian; numerical adaptivity; advection–diffusion equations; stabilized Lagrangian; advection-diffusion equations.

UR - http://eudml.org/doc/245174

ER -

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