Optimal control and numerical adaptivity for advection-diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 5, pp. 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
Mots clés : optimal control problems, partial differential equations, finite element approximation, stabilized lagrangian, numerical adaptivity, advection-diffusion equations
     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},
     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/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, Tome 39 (2005) no. 5, pp. 1019-1040. doi : 10.1051/m2an:2005044. http://www.numdam.org/item/M2AN_2005__39_5_1019_0/

[1] V.I. Agoshkov, Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Science, Moscow (2003).

[2] A.K. Aziz, J.W. Wingate and M.J. Balas, Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York (1971). | MR 428151 | Zbl 0409.00018

[3] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | Zbl 1105.65349

[4] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concepts. SIAM J. Control Optim. 39 (2000) 113-132. | Zbl 0967.65080

[5] M. Braack and A. Ern, A posteriori control of modelling errors and Discretization errors. SIAM Multiscale Model. Simul. 1 (2003) 221-238. | Zbl 1050.65100

[6] G. Finzi, G. Pirovano and M. Volta, Gestione della Qualità dell'aria. Modelli di Simulazione e Previsione. Mc Graw-Hill, Milano (2001).

[7] L. Formaggia, S. Micheletti and S. Perotto, 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 1107.65098

[8] A.N. Kolmogorov and S.V. Fomin, Elements of Theory of Functions and Functional Analysis. V.M. Tikhomirov, Nauka, Moscow (1989). | MR 1025126 | Zbl 0672.46001

[9] R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distribuited elliptic optimal control problems. SIAM J. Control Optim. 41 (2001) 1321-1349. | Zbl 1034.49031

[10] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971). | MR 271512 | Zbl 0203.09001

[11] W. Liu and N. Yan, A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math. 120 (2000) 159-173. | Zbl 0963.65072

[12] W. Liu and N. Yan, A Posteriori error estimates for distribuited convex optimal control problems. Adv. Comput. Math. 15 (2001) 285-309. | Zbl 1008.49024

[13] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001). | MR 1835648 | Zbl 0970.76003

[14] M. Picasso, Anisotropic a posteriori error estimates for an optimal control problem governed by the heat equation. Int. J. Numer. Method PDE (2004), submitted. | Zbl 1111.65060

[15] O. Pironneau and E. Polak, Consistent approximation and approximate functions and gradients in optimal control. SIAM J. Control Optim. 41 (2002) 487-510. | Zbl 1011.49020

[16] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin and Heidelberg (1994). | MR 1299729 | Zbl 0803.65088

[17] J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization (Shape Sensitivity Analysis). Springer-Verlag, New York (1991). | MR 1215733 | Zbl 0761.73003

[18] R.B. Stull, An Introduction to Boundary Layer Meteorology. Kluver Academic Publishers, Dordrecht (1988). | Zbl 0752.76001

[19] F.P. Vasiliev, Methods for Solving the Extremum Problems. Nauka, Moscow (1981).

[20] D.A. Venditti and D.L. Darmofal, Grid adaption for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 176 (2002) 40-69. | Zbl 1120.76342

[21] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996). | Zbl 0853.65108