Error Estimates for Finite Element Approximation of Dirichlet Boundary Control for Stokes Equations in L2(Γ)

In this paper we study the finite element approximation of Dirichlet boundary control for Stokes equations in the control space L-2(Gamma). The governing state equation is understood in the very weak sense which ensures the well-posedness of the optimization problem. We consider the boundary control problems posed on either convex polygonal domains or smooth domains, and derive the respective regularity results for the solutions. A priori error estimates for the Taylor-Hood finite element approximation to the optimization problem are derived for the control variable for both classes of computational domains, where the smooth domain is approximated by a polygonal one. We obtain nearly half an order convergence for the control on convex polygonal domains that reflects the global regularity of the control, while in smooth domains we obtain first order convergence that is dominated by the geometric approximation error of the outward normal vector. Numerical experiments are presented to validate the theoretical results.