A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin method to approximate the solution. We use polynomials of degree to approximate the state and dual state, and polynomials of degree to approximate their fluxes. Moreover, we use polynomials of degree k to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when . Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when . We illustrate our convergence results with numerical experiments.