Coupling stabilized finite element methods with finite difference time integration for advection-diffusion-reaction problems

We present some numerical schemes for the unsteady advection–diffusion–reaction linear problem, in one space dimension. We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged). Discretization in time is performed by using the Crank–Nicolson finite difference scheme, while for the space discretization we consider three classical stabilized finite element schemes and the more recent Link-Cutting Bubble strategy proposed in [F. Brezzi, G. Hauke, L.D. Marini, G. Sangalli, Link-cutting bubbles for the stabilization of convection–diffusion–reaction problems, Math. Models Methods Appl. Sci. 13 (2003) 445–461]. Numerical experiments are presented to assess and valuate the capabilities of the proposed methods. An L1-error analysis of the Link-Cutting Bubble strategy for solving the steady problem is included.