The Effect of Stabilization in Finite Element Methods
for the Optimal Boundary Control of the Oseen Equations
Feby Abraham and Marek Behr
Department of Mechanical Engineering and Materials Science
Matthias Heinkenschloss
Department of Computational and Applied Mathematics
Rice University
Finite Elements in Analysis and Design, Vol. 41 (2004),
pp. 229-251.
Abstract
We study the effect of the Galerkin/Least-Squares (GLS)
stabilization on the finite element
discretization of optimal control problems
governed by the linear Oseen equations.
Control is applied in the form of suction or blowing on part
of the boundary.
Two ways of including the GLS stabilization into the discretization
of the optimal control problem are discussed.
In one case the optimal control problem is first discretized
and the resulting finite dimensional problem is then solved.
In the other case, the optimality contitions are first formulated
on the differential equation level and are then discretized.
Both approaches lead to different discrete adjoint equations
and, depending on the choice of the stabilization parameters
and grid size, may significantly affect the computed control.
The effect of the order in which the discretization is applied
and the choice of the
stabilization parameters are illustrated using two test problems.
The cause of the differences in the computed controls are
explored numerically.
Diagnostics are introduced that may guide the selection of
sensible stabilization parameters.