Discontinuous Galerkin Approximations for Distributed Optimal Control Problems Constrained by Parabolic PDE's

Abstract

A discontinuous Galerkin finite element method for optimal control problems having states constrained by linear parabolic PDE's is examined. The spacial operator may depend on time and need not be self-adjoint. The schemes considered here are discontinuous in time but conforming in space. Fully-discrete error estimates of arbitrary order are presented and various constants are tracked. In particular, the estimates are valid for small values of $\alpha$, $\gamma$, where $\alpha$ denotes the penalty parameter of the cost functional and $\gamma$  the coercivity constant. Finally, error estimates for the convection dominated convection-diffusion equation are presented, based on a Lagrangian moving mesh approach.