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Two difierent approaches are proposed to enhance the e-ciency of the nu- merical resolution of optimal control problems governed by a linear advection{ difiusion 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 numer- ical adaptive strategy for the generation of e-cient 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 numer- ical tests show that by this technique a substantial saving of computational costs can be achieved.

Numerical Approximation of a Control Problem for Advection-Diffusion Processes