This short article illustrates some experiences in implementing a discontinuity capturing technique in the wider context of a generalised Taylor-Galerkin/pressure correction scheme for viscoelastic problems. In a companion study of Carew et al., to be read in conjunction, a detailed investigation has been conducted into the implementation of a consistent Taylor-streamline upwind/Petrov-Galerkin (TSUPG) algorithm for simulating flow through a 4:1 planar contraction. This approach involves a time-stepping scheme and decoupled equations on each time step. The purpose of this note is to indicate the effect that discontinuity capturing has on numerical convergence to steady solutions for Oldroyd-B fluids at high levels of elasticity. Under such circumstances, these techniques are found to im rove stability properties but not convergence to a steady state.