Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

M. Heinkenschloss
Department of Computational and Applied Mathematics
Rice University

D. Leykekhman
Department of Mathematics
University of Connecticut

SIAM Journal on Numerical Analysis, Vol. 47, No. 6 (2010) pp. 4607-4638


We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved.
We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is of first order even if the error is computed locally in a region away from the boundary layer where the exact solution is smooth. We also prove optimal weighted error estimates. These imply optimal convergence rates for the local error in regions away from interior layers. Numerical examples are presented to illustrate some of the theoretical results.


Optimal control, advection-diffusion equations, discretization, local error estimates, stabilized finite elements.