Non-Newtonian flow has a nonlinear constitutive relationship with an advective nature. It was found that in highly advective (convective) problems, the Galerkin formulation "under-diffused," resulting in divergence at low elastic numbers. The use of the Streamline-Upwinding (SU) method improved the solution, especially when used with the Phan-Thien-Tanner (PTT) model. At the boundary discontinuity, however, the stress gradient did not necessarily flow along the streamline direction, and oscillations still remained at the corner. The Discontinuity-Capturing (DC) method resolved this problem by applying control in the direction of the stress gradient rather than the stream line direction, and a smoother solution at the corner region was achieved.