Numerical Approximation of a Control Problem for Advection-Diffusion Processes

Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection– diffusion equation. In the framework of the Galerkin–Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of input–output relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.