Employing a finite element computing scheme implemented onto the Leonov viscoelastic model, we describe various kinds of melt fracture for the extrudate exiting from the Poiseuille flow in the straight and contraction channels with wall slip ignored. Four types such as sharkskin, gross melt fracture, slow surface undulation and ripples are found depending on the flow conditions like the flowrate, geometry and liquid property, and they are expressed as 2D elastic instability in this inertialess flow regime. Even though not considered, the effect of die wall slip has to be included in the realistic modeling of melt fracture. However, here we make the first attempt to analyzing extrudate instability in terms of purely fluid mechanical factors. As a result, each type of melt fracture is verified to result from the different origin, and thus geometric singularities and streamline vortices at contraction corner and die exit determine this type of extrudate distortion. Keywords: Melt fracture, Sharkskin, Numerical simulation, Leonov model, Elastic instability