Numerical simulations for the benchmark extrusion problem of an IUPAC-LDPE melt through capillary dies of infinite length have been performed using the finite element method with the u-v-p formulation. A modified version of the K-BKZ integral constitutive equation (the PSM model) has been used to model the viscoelastic stresses of the melt. Mesh refinement with five different meshes is carried out to check both mesh-dependency and convergence of the numerical scheme. The present simulations reveal that monotonic convergence to machine accuracy is obtained for all variables, except for those on the first free surface node which oscillate around a small mean value. It is also found that mesh-independent results can be obtained for the swelling ratio and exit correction with relatively dense meshes for the whole range of stable experimental conditions before melt fracture occurs. However, loss of convergence occurs for lower values of the apparent shear rate when severe discretization is applied around the singular point of the domain at the die exit.