Time domain decomposition of parabolic control problems based on discontinuous Galerkin semi-discretization

A parallel-in-time method for the optimal control problem governed by linear parabolic equations is analyzed. The corresponding optimal system consists of a state equation (forward in time) and an adjoint equation (backward in time) which are coupled by the necessary condition. The whole time interval is divided into non-overlapping subdomains which are associated by the Robin-type transmission conditions. We apply discontinuous Galerkin method for time discretization in each subdomain and keep continuous in space. In order to prove the convergence of the algorithm, the semi-discretization errors at initial and final points are obtained based on nodal stability estimates. We complete the convergence analysis for iterative solution of the nonoverlapping Schwarz method through energy estimates for two subdomains and present the convergence rate affected by the transmission parameters. Finally, numerical experiments verify the theoretical results.