Petrov-galerkin crank-nicolson scheme for parabolic optimal control problems on nonsmooth domains

In this paper we transfer the a priori error analysis for the discretization of parabolic optimal control problems on domains allowing for H² regularity (i.e. either with smooth boundary or polygonal and convex) to a large class of nonsmooth domains. We show that a combination of two ingredients for the optimal convergence rates with respect to the spatial and the temporal discretization is required. First we need a time discretization scheme which has the desired convergence rate in the smooth case. Secondly we need a method to treat the singularities due to non-smoothness of the domain for the corresponding elliptic state equation. In particular we demonstrate this philosophy with a Crank-Nicolson time discretization and finite elements on suitably graded meshes for the spatial discretization. A numerical example illustrates the predicted convergence rates.