The 1-D convection diffusion equation: Galerkin least squares approximations and feedback control

The standard Galerkin finite element approximation of the convection diffusion equation is known to be numerically unstable for small values of the diffusion parameter. One way to overcome this difficulty is to use a stabilized finite element method, such as Galerkin least squares, and one finds this approach in the simulation literature. In this paper, we investigate the effect of a stabilized finite element approximation of the convection diffusion equation in the context of feedback control design. The issue at hand is how the additional stabilizing terms affect the resulting controller. We show that the stabilized system provides accurate controllers, and can compute them on coarser grids than the unstabilized system.