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Analysis of the Streamline Upwind/Petrov Galerkin Method

Applied to the Solution of Optimal Control Problems

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S. Scott Collis

Department of Mechanical Engineering and Materials Science

Matthias Heinkenschloss

Department of Computational and Applied Mathematics

Rice University

CAAM Technical Report TR02-01

March 2002.

### Abstract

We study the effect of the streamline upwind/Petrov Galerkin (SUPG)
stabilized finite element method on the discretization of optimal
control problems governed by linear advection-diffusion equations.
We compare two approaches for the numerical solution of such
optimal control problems.
In the discretize-then-optimize approach the optimal control problem
is first discretized,
using the SUPG method for the discretization of the advection-diffusion
equation, and then the resulting finite dimensional optimization
problem is solved.
In the optimize-then-discretize approach one first computes the infinite
dimensional
optimality system, involving the advection-diffusion equation as well
as the adjoint advection-diffusion equation, and then discretizes
this optimality system using the SUPG method for both the
original and the adjoint equations.
These approaches lead to different results. The main result of this paper
is an estimates for the error between
the solution of the infinite dimensional optimal control problem
and their approximations computed using the previous approaches.
For a class of problems prove that the optimize-then-discretize approach
has better asymptotic convergence properties if finite elements of order
greater than one are used. For linear finite elements our theoretical
convergence results for both approaches are comparable, except in the
zero diffusion limit where again the optimize-then-discretize approach
seems favorable.
Numerical examples are presented to illustrate some of the theoretical
results.
PDF file (250 KB).