On fully implicit spce-time discretization for motions of incompressible fluids with shear-dependent viscosities: The case p <= 2

In this paper, we deal with a rigorous error analysis for a fully implicit space-time discretization of an unsteady, non-Newtonian fluid flow model, w here the nonlinear operator related to the stress tensor exhibits p-structu re. In a first step, a semidiscretization in time using the implicit Euler method is discussed. Due to limitation of regularity of the solution for th e case p not equal 2, a decrease with respect to convergence rates of the m ethod is stated, in general, by retaining smoothly rst order in appropriate norms for the Stokes law ( i.e., p = 2). The analysis is then extended to a full discretization using stable pairings of finite element spaces in a s econd step.