A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

TL;DR

This paper introduces a superconvergent hybridizable discontinuous Galerkin (HDG) method for efficiently solving distributed control problems governed by convection-diffusion PDEs, achieving optimal error estimates and superconvergence.

Contribution

The paper proposes a novel HDG method with polynomial degrees for state, dual state, and fluxes, providing optimal error bounds and superconvergence for control approximation.

Findings

Optimal a priori error estimates for all variables when k > 0

Superconvergence achieved for state, dual state, and control when k ≥ 1

Numerical experiments confirm theoretical convergence results

Abstract

We consider a distributed optimal control problem governed by an elliptic convection 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 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 $k\geq 1$. We illustrate our convergence results with numerical experiments.