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

摘要

This paper analyzes the local properties of the symmetric interior penalty upwinddiscontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governedby linear reaction-advection-diffusion equations with distributed controls. The theoretical andnumerical results presented in this paper show that for advection-dominated problems the convergenceproperties of the SIPG discretization can be superior to the convergence properties of stabilizedfinite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. Forexample, we show that for a small diffusion parameter the SIPG method is optimal in the interiorof the domain. This is in sharp contrast to SUPG discretizations, for which it is known that theexistence of boundary layers can pollute the numerical solution of optimal control problems everywhereeven into domains where the solution is smooth and, as a consequence, in general reducesthe convergence rates to only first order. In order to prove the nice convergence properties of theSIPG discretization for optimal control problems, we first improve local error estimates of the SIPGdiscretization for single advection-dominated equations by showing that the size of the numericalboundary 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 intodomains of smoothness in optimal control problems. This favorable property of the SIPG method isdue 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. Theimportance of the weak treatment of boundary conditions for the solution of advection dominatedoptimal control problems with distributed controls is also supported by our numerical results.