# Optimal control and numerical adaptivity for advection–diffusion equations

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 39.5 (2010): 1019-1040. <http://eudml.org/doc/194289>.

@article{Dede2010,

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},

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

language = {eng},

month = {3},

number = {5},

pages = {1019-1040},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/194289},

volume = {39},

year = {2010},

}

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

DA - 2010/3//

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.; optimal control problems; partial differential equations; stabilized Lagrangian; numerical adaptivity; advection-diffusion equations.

UR - http://eudml.org/doc/194289

ER -

## References

top- 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).
- R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer.10 (2001) 1–102.
- 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.
- M. Braack and A. Ern, A posteriori control of modelling errors and Discretization errors. SIAM Multiscale Model. Simul.1 (2003) 221–238.
- G. Finzi, G. Pirovano and M. Volta, Gestione della Qualità dell'aria. Modelli di Simulazione e Previsione. Mc Graw-Hill, Milano (2001).
- 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.
- A.N. Kolmogorov and S.V. Fomin, Elements of Theory of Functions and Functional Analysis. V.M. Tikhomirov, Nauka, Moscow (1989).
- 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.
- J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971).
- W. Liu and N. Yan, A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math.120 (2000) 159–173.
- W. Liu and N. Yan, A Posteriori error estimates for distribuited convex optimal control problems. Adv. Comput. Math.15 (2001) 285–309.
- 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.
- O. Pironneau and E. Polak, Consistent approximation and approximate functions and gradients in optimal control. SIAM J. Control Optim.41 (2002) 487–510.
- 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).
- 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.
- R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996).

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