Numerical solutions are obtained for the 2D confined plane viscoelastic flow around a confined cylinder. The Navier-Stokes momentum differential fluids, whilst for the viscoelastic case different constitutive equations are utilised.Finite element scheme is implemented to represent the convective terms in the constitutive equations for the Phan-Thien-Tanner fluids, and the resulting predictions of the drag coefficient on the cylinder are shown to be as accurate as existing numerical predictions. In this context, a Taylor-Galerkin/pressure correction finite element method has been adopted for the parabolic-elliptic momentum-continuity equations, whilst a finite volume implementation is utilised for the hyperbolic sub-system comprising of the constitutive equation..