Analysis of the SUPG Method for the Solution of Optimal Control Problems

TL;DR

This paper compares two numerical approaches for solving optimal control problems governed by advection-diffusion equations using the SUPG finite element method, analyzing their error estimates and convergence properties.

Contribution

It provides theoretical error estimates and convergence analysis for both discretize-then-optimize and optimize-then-discretize approaches with SUPG, highlighting their differences.

Findings

Optimize-then-discretize has better asymptotic convergence with higher-order elements.

For linear elements, convergence is similar except in zero diffusion limit.

Numerical examples confirm theoretical error estimates.

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 are 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.