A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs I: High Regularity

TL;DR

This paper introduces a novel HDG method for Dirichlet boundary control problems governed by convection diffusion PDEs, providing theoretical analysis and optimal error estimates, with numerical validation.

Contribution

The paper develops a new HDG approach for convection diffusion Dirichlet control problems, including well-posedness, regularity results, and optimal error estimates in 2D.

Findings

Established well-posedness and regularity for the control problem

Derived optimal a priori error estimates in 2D

Numerical experiments confirm the effectiveness of the HDG method

Abstract

We propose a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a Dirichlet boundary control problem governed by an elliptic convection diffusion PDE. Even without a convection term, Dirichlet boundary control problems are well-known to be very challenging theoretically and numerically. Although there are many works in the literature on Dirichlet boundary control problems for the Poisson equation, the authors are not aware of any existing theoretical or numerical analysis works for convection diffusion Dirichlet control problems. We make two contributions. First, we obtain well-posedness and regularity results for the Dirichlet control problem. Second, under certain assumptions on the domain and the target state, we obtain optimal a priori error estimates in 2D for the control for the new HDG method. As far as the authors are aware, there are no existing comparable results in the literature. We present numerical experiments to demonstrate the performance of the HDG method.