Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:06:27.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal control and numerical adaptivityfor advection–diffusionequations

Published online by Cambridge University Press:  15 September 2005

Luca Dede'  and
Alfio Quarteroni
Luca Dede'
Affiliation:
MOX–Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy. luca.dede@mate.polimi.it
Alfio Quarteroni
Affiliation:
MOX–Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy. luca.dede@mate.polimi.it École Polytechnique Fédérale de Lausanne (EPFL), FSB, Chaire de Modelisation et Calcul Scientifique (CMCS), Station 8, 1015, Lausanne, Switzerland. alfio.quarteroni@epfl.ch
Get access

Abstract

We propose a general approach for the numerical approximation ofoptimal control problems governed by a linear advection–diffusionequation, based on a stabilization method applied to theLagrangian functional, rather than stabilizing the state andadjoint equations separately. This approach yields a coherentlystabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order termsare needless. Our a posteriori estimates stems from splitting theerror on the cost functional into the sum of an iteration errorplus a discretization error. Once the former is reduced below agiven threshold (and therefore the computed solution is “near”the optimal solution), the adaptive strategy is operated on thediscretization error. To prove the effectiveness of the proposedmethods, we report some numerical tests, referring to problems inwhich the control term is the source term of theadvection–diffusion equation.

Keywords

Optimal control problemspartial differentialequationsfinite element approximationstabilizedLagrangiannumerical adaptivityadvection–diffusion equations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 1019 - 1040
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

V.I. Agoshkov, Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Science, Moscow (2003).
A.K. Aziz, J.W. Wingate and M.J. Balas, Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York (1971).
Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. CrossRef
Becker, R., Kapp, H. and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: basic concepts. SIAM J. Control Optim. 39 (2000) 113–132. CrossRef
Braack, M. and Ern, A., A posteriori control of modelling errors and Discretization errors. SIAM Multiscale Model. Simul. 1 (2003) 221–238. CrossRef
G. Finzi, G. Pirovano and M. Volta, Gestione della Qualità dell'aria. Modelli di Simulazione e Previsione. Mc Graw-Hill, Milano (2001).
Formaggia, L., Micheletti, S. and Perotto, S., Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math. 51 (2004) 511–533. CrossRef
A.N. Kolmogorov and S.V. Fomin, Elements of Theory of Functions and Functional Analysis. V.M. Tikhomirov, Nauka, Moscow (1989).
Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation for distribuited elliptic optimal control problems. SIAM J. Control Optim. 41 (2001) 1321–1349. CrossRef
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971).
Liu, W. and Yan, N., A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math. 120 (2000) 159–173. CrossRef
Liu, W. and Yan, N., A Posteriori error estimates for distribuited convex optimal control problems. Adv. Comput. Math. 15 (2001) 285–309. CrossRef
B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001).
M. Picasso, Anisotropic a posteriori error estimates for an optimal control problem governed by the heat equation. Int. J. Numer. Method PDE (2004), submitted.
Pironneau, O. and Polak, E., Consistent approximation and approximate functions and gradients in optimal control. SIAM J. Control Optim. 41 (2002) 487–510. CrossRef
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin and Heidelberg (1994).
J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization (Shape Sensitivity Analysis). Springer-Verlag, New York (1991).
R.B. Stull, An Introduction to Boundary Layer Meteorology. Kluver Academic Publishers, Dordrecht (1988).
F.P. Vasiliev, Methods for Solving the Extremum Problems. Nauka, Moscow (1981).
Venditti, D.A. and Darmofal, D.L., Grid adaption for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 176 (2002) 40–69. CrossRef
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996).