Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems

摘要

The aim of this paper is to investigate commutative properties of a large family of discontinuous Galerkin (DG) methods applied to optimal control problems governed by the advection-diffusion equations. To compute numerical solutions of PDE constrained optimal control problems there are two main approaches: optimize-then-discretize and discretize-then-optimize. These two approaches do not always coincide and may lead to substantially different numerical solutions. The methods for which these two approaches do coincide we call commutative. In the theory of single equations, there is a related notion of adjoint or dual consistency. In this paper we classify DG methods both in primary and mixed forms and derive necessary conditions that can be used to develop new commutative methods. We will also derive error estimates in the energy and L~2 norms. Numerical examples reveal that in the context of PDE constrained optimal control problems a special care needs to be taken to compute the solutions. For example, choosing non-commutative methods and discretize-then-optimize approach may result in a badly behaved numerical solution.