Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

Abstract

This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advection-dominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advection-dominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results.