A Superconvergent Hybridizable Discontinuous Galerkin Method for Dirichlet Boundary Control of Elliptic PDEs
It provides a rigorous numerical analysis and optimal convergence for a challenging class of PDE-constrained optimization problems with low-regularity solutions.
This paper develops a hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs, achieving optimal a priori error estimates and superlinear convergence for the control in 2D convex polygonal domains, validated by numerical experiments.
We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.