A new transient 3D finite element method for predicting extrudate swell of domains containing sharp edges is proposed. Here, the sharp edge is maintained over a large distance in the extrudate by describing the corner lines as material lines. The positions of these lines can be used to describe the transverse swelling of the 2D free surfaces and expand the domain over which a 2D height function on the free surfaces is applied. Solving the 2D height functions gives the positions of the free surfaces. First a 2D axisymmetric case was tested for comparison, using three different constitutive models. The Giesekus, linear Phan-Thien Tanner (PTT) and exponential PTT constitutive models all showed convergence upon mesh- and time-step refinement. It was found that convergence remains challenging due to the singularity at the die exit. The new method is validated by comparing the final volume change of the extrudate of a 3D cylinder to the final volume change of a reference mesh of the 2D axisymmetric case. Finally, simulations were performed for different, complex, die shapes for a viscous fluid and for viscoelastic fluids. The results compared favorably with literature. Viscoelastic results, using the Giesekus model and the exponential PTT model, were compared for different Weissenberg numbers and different values for the non-linear parameters of the constitutive models. It was found that the swell is highly dependent on the rheological parameters and the constitutive model used.