Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

¹ NCMIS & LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China
² Department of Mathematics, University of Georgia, Athens, GA, USA
³ Dpto. de Matemáticas. Universidad de Oviedo Campus de Gijón, Gijón, Spain
⁴ Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA
⁵ Department of Mathematics Science, University of Delaware, Newark, DE, USA

^* Corresponding author: singlerj@mst.edu

Received: 24 August 2019
Accepted: 3 March 2020

Abstract

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L² penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.