Steady creeping and inertial flow of shear thinning viscoelastic fluids past periodic square arrays of cylinders is simulated using a higher order Galerkin finite element method with a variety of nonlinear differential constitutive equations. The results obtained with all the viscoelastic models show that as the pressure drop is progressively increased the flow resistance decreases initially, goes through a minimum and then asymptotes to its initial value. The computed results are compared with earlier measurements of Skartsis et al. with a 2% aqueous polyacrylamide solution. It is shown that the experimental and theoretical findings produce similar results below a critical Reynolds or Weissenberg number. Above this critical value the experimental results show a dramatic increase in flow resistance while the computational results show a reduction in flow resistance. This discrepancy between the experimental and computational results is due to the fact that at a critical Weissenberg number the steady base flow exhibits a purely elastic nonlinear transition to a time dependent flow structure that cannot be modeled by two-dimensional steady numerical simulations.