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 k + 1 to approximate the state and dual state, and polynomials of degree k ≥ 0 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 k ≥ 0. 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 k ≥ 1. We illustrate our convergence results with numerical experiments