A Galerkin finite element method is used to solve the extrudate swell problem for both a convected Maxwell fluid, and for a fluid with a nonzero second normal stress difference. Exit pressure calculations are undertaken to investigate the validity of the theoretical formulations relating exit pressures to normal stress differences for these two fluids. For the particular fluids and range of shear stresses considered, the numerical results indicate: (1) The underlying incorrect assumption of viscometric flow to the die exit results in normal stress values which are too high. However, this effect is not significant enough to invalidate exit pressure measurements as a means of estimating normal stresses. (2) The exit pressure theory is sometimes applied using the assumptionPc*=0.5 p(0,L)c(orPc*=Ps*) since Han and co‐workers find experimentally thatPc*=Ps*.We find for the convected Maxwell model(N2/N1=0.0)thatPc*=p(0,L)cwhereas for a model withN2/N1=−0.2,Pc*=0.84 p(0,L)c.Our calculations suggest, at least, that the equivalence (or not) ofPc*andPs*depends onN2/N1.(3) For the convected Maxwell fluid:Pc*/τw=0.35(N1/τw)+0.55for a capillary die,Ps*/τw=0.28(N1/τw)+0.30for a slit die. The latter is in agreement with the slit die results of Reddy and Tanner.