Abstract: Standard Galerkin finite element discretizations applied to advection-dominated, elliptic PDEs can lead to highly oscillatory solutions, unless the grid is sufficiently fine. Over the years a number of stabilized methods, such as streamline upwind/Petrov Galerkin (SUPG) methods or Discontinuous Galerkin (DG) methods, were developed. These methods are frequently applied to advection-dominated elliptic PDEs. Local and global error estimates for these methods are well known.
In this talk we will address several issues that are specific to finite element approximation of optimal control problems. First, we will discuss two discretization strategies: discretize-then-optimize and optimize-then-discretize. It turns out that for many stabilized methods these two approaches are not equivalent and may lead to very different numerical solutions. Then, we will concentrate on local and global error estimates for optimal control problems. We will show that although the global error estimates for a single advection-diffusion equation and optimal control problems are similar, the local error estimates are very different. For such error estimates it is essential how the boundary conditions are enforced. We will present numerical results to illustrate our findings.