Crank–Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection–diffusion equations

We consider a finite element method with symmetric stabilization for transient advection-diffusion-reaction problems. The Crank-Nicolson finite difference scheme is used for discretization in time. We prove stability of the numerical method both for implicit and explicit treatment of the stabilization operator. The resulting convergence results are given and the results are illustrated by a numerical experiment. We then consider a model problem for pde-constrained optimization. Using discrete adjoint consistency of our stabilized method we show that both the implicit and semi-implicit methods proposed yield optimal convergence for the control and the state variable.