Finite element methods with symmetric stabilization for the transient convection–diffusion–reaction equation

We consider implicit and semi-implicit time-stepping methods for finite element approximations of singularly perturbed parabolic problems or hyperbolic problems. We are interested in problems where the advection dominates and stability is obtained using a symmetric, weakly consistent stabilization operator in the finite element method. Several AA-stable time discretizations are analyzed and shown to lead to unconditionally stable and optimally convergent schemes. In particular, we show that the contribution from the stabilization leading to an extended matrix pattern may be extrapolated from previous time steps, and hence handled explicitly without loss of stability and accuracy. A fully explicit treatment of the stabilization term is obtained under a CFL condition.