A stable and accurate implementation is presented of a cell-vertex hybrid finite volume/element scheme based upon triangular meshes. This scheme is novel to this domain and is applied to the numerical solution of Oldroyd model fluids in abrupt planar contraction flows. All important has been the use of non-conservative flux representation, consistency in treatment of transients, fluxes and sources, and a recovery technique for velocity gradients. Linear stress representation, with non-recovered stress gradients, has proved crucial to widening stability thresholds. Solution smoothness at the higher levels of elasticity results, so that converged solutions are attainable. We have also highlighted the importance of incorporating a reduced-integration local discontinuity capturing technique for the re-entrant corner solution. A diminishing lip vortex with mesh refinement is reported. With increasing elasticity, lip vortex growth and a diminishing salient corner vortex is noted. In addition, a trailing-edge vortex is found to accompany the onset of a lip vortex. Agreement with analytical theory is observed through the asymptotic behaviour of velocities and stresses near the re-entrant corner, a region where the loss of evolution of the discrete system is investigated.