Optimal Control and Numerical Adaptivity for Advection–diffusion Equations

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. Mathematics Subject Classification. 35J25, 49J20, 65N30, 76R50. Received: February 9, 2005. Revised: April 22, 2005. Introduction Many physical problems can be modelled by linear advection–diffusion partial differential equations; this is the case for example if we want to forecast the distribution of a substance in a continuous medium, such as a pollutant in air or water. In this contest it is interesting to operate the source terms (e.g. the emission rate of pollutants) in order that the PDE solution approaches as closely as possible a desired distribution (or, otherwise said, the concentration of pollutant stands below a pre-assigned threshold). This aspect can be conveniently accommodated in the framework of the optimal control theory, for which we assume as control function the source term, while the “observation” is a function depending on the PDE solution. The classical approach to this kind of problems is based on the theory developed by Lions [10] (see also [2,13,15,17]), or, as complementary to the previous one, on the Lagrangian functional formalism [4]. By adopting the latter methodology, which is useful for practical problems, but it does not ensure the existence and uniqueness of solution (see [1]), we address a generic optimal control problem applied to an advection–diffusion equation. For its approximation we use an iterative method applied to the Galerkin-FE discretization of both state and adjoint equations. To get rid of numerical instabilities arising in the transport dominated regimes, we propose a stabilization on the Lagrangian