A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

Abstract

We investigated a hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work (Gong et al. SIAM J Numer Anal 56(4):2262–2287, 2018) and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen et al. (J Comput Phys 302:674–692, 2015) for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results.