A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity
Provides theoretical guarantees for a numerical method in PDE-constrained optimization under minimal regularity, advancing computational control of convection-diffusion problems.
The paper extends convergence analysis of an HDG method for Dirichlet boundary control of convection-diffusion PDEs to low-regularity settings, proving optimal convergence rates without previous restrictive assumptions.
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.