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