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We consider a distributed optimal control problem governed by an ellipticconvection diffusion PDE, and propose a hybridizable discontinuous Galerkin(HDG) method to approximate the solution. We use polynomials of degree $k+1$and $k \ge 0$ to approximate the state, dual state, and their fluxes,respectively. Moreover, we use polynomials of degree $k$ to approximate thenumerical traces of the state and dual state on the faces, which are the onlyglobally coupled unknowns. We prove optimal a priori error estimates for allvariables when $ k > 0 $. Furthermore, from the point of view of the number ofdegrees of freedom of the globally coupled unknowns, this method achievessuperconvergence for the state, dual state, and control when $k\geq 1$. Weillustrate our convergence results with numerical experiments.

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs