In this work, we study the penalty finite element approximation of the stationary
power law Stokes problem. We prove uniform convergence of the finite element
solution with respect to the penalized parameter under classical assumptions on the
weak solution. We formulate and analyze the convergence of a nonlinear saddle point
problem by adopting a particular algorithm based on vanishing viscosity approach
and long time behavior of an initial value problem. Finally, the predictions observed
theoretically are validated by means of numerical experiments.