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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an:2005044/
[1] , Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Science, Moscow (2003).
[2] , and, Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York (1971). | MR | Zbl
[3] and, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | Zbl
[4] , and, Adaptive finite element methods for optimal control of partial differential equations: basic concepts. SIAM J. Control Optim. 39 (2000) 113-132. | Zbl
[5] and, A posteriori control of modelling errors and Discretization errors. SIAM Multiscale Model. Simul. 1 (2003) 221-238. | Zbl
[6] , and, Gestione della Qualità dell'aria. Modelli di Simulazione e Previsione. Mc Graw-Hill, Milano (2001).
[7] , and, Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math. 51 (2004) 511-533. | Zbl
[8] and, Elements of Theory of Functions and Functional Analysis. V.M. Tikhomirov, Nauka, Moscow (1989). | MR | Zbl
[9] ,, and, Adaptive finite element approximation for distribuited elliptic optimal control problems. SIAM J. Control Optim. 41 (2001) 1321-1349. | Zbl
[10] , Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971). | MR | Zbl
[11] and, A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math. 120 (2000) 159-173. | Zbl
[12] and, A Posteriori error estimates for distribuited convex optimal control problems. Adv. Comput. Math. 15 (2001) 285-309. | Zbl
[13] and, Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001). | MR | Zbl
[14] , Anisotropic a posteriori error estimates for an optimal control problem governed by the heat equation. Int. J. Numer. Method PDE (2004), submitted. | Zbl
[15] and, Consistent approximation and approximate functions and gradients in optimal control. SIAM J. Control Optim. 41 (2002) 487-510. | Zbl
[16] and, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin and Heidelberg (1994). | MR | Zbl
[17] and, Introduction to Shape Optimization (Shape Sensitivity Analysis). Springer-Verlag, New York (1991). | MR | Zbl
[18] , An Introduction to Boundary Layer Meteorology. Kluver Academic Publishers, Dordrecht (1988). | Zbl
[19] , Methods for Solving the Extremum Problems. Nauka, Moscow (1981).
[20] and, Grid adaption for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 176 (2002) 40-69. | Zbl
[21] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996). | Zbl
Cité par Sources :
