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.

翻译：本文研究了流线迎风/Petrov-Galerkin（SUPG）稳定化有限元方法对线性对流扩散方程控制的最优控制问题离散化的影响。我们比较了此类最优控制问题的两种数值求解方法：在"先离散后优化"方法中，首先采用SUPG方法离散对流扩散方程，对最优控制问题进行离散化，然后求解得到的有限维优化问题；而在"先优化后离散"方法中，首先推导包含对流扩散方程及其伴随方程的无限维最优性系统，随后采用SUPG方法对原始方程和伴随方程进行离散化。这两种方法会产生不同的计算结果。本文的主要成果是给出了无限维最优控制问题解与通过上述方法计算的近似解之间的误差估计。针对一类特定问题，我们证明当采用高于一阶的有限元时，"先优化后离散"方法具有更优的渐近收敛特性。对于线性有限元，两种方法的理论收敛结果基本相当，但在扩散系数趋于零的极限情况下，"先优化后离散"方法再次显现优势。文中通过数值算例验证了部分理论结果。