Finite-element approximation of viscoelastic fluid flow with slip boundary condition

To treat the discrete case, we assume that the continuous solution of the non-Newtonian problem exists and is small and smooth. Approximating the extra stress, velocity, pressure, and normal viscoelastic stress component via P1 discontinuous, P2 continuous, P1 continuous, and P0 discontinuous elements, respectively, yields a stable finite-element scheme. Finally, via a fixed point argument, we establish the existence of an approximate solution and derive error estimates.