A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity

TL;DR

This paper develops a new HDG method for Dirichlet boundary control of convection-diffusion PDEs, achieving optimal convergence rates even with low regularity solutions, supported by theoretical analysis and numerical validation.

Contribution

It extends previous work by providing convergence analysis without regularity assumptions, demonstrating optimal control approximation for low regularity solutions.

Findings

Proved optimal convergence rates for low regularity solutions.

Validated theoretical results with numerical experiments.

Extended HDG method applicability to more general cases.

Abstract

In the first part of this work, we analyzed a Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity and we use a different analysis approach. We again prove an optimal convergence rate for the control, and present numerical results to illustrate the convergence theory.