Optimal control and numerical adaptivity for advection–diffusion equations

¹ MOX–Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, 20133, Milano, Italy. luca.dede@mate.polimi.it
² École Polytechnique Fédérale de Lausanne (EPFL), FSB, Chaire de Modelisation et Calcul Scientifique (CMCS), Station 8, 1015, Lausanne, Switzerland. alfio.quarteroni@epfl.ch

Received: 9 February 2005
Revised: 22 April 2005

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.