On Fully Implicit Space-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, where the nonlinear operator related to the stress tensor exhibits p-structure. In a first step, a semidiscretization in time using the implicit Euler method is discussed. Due to limitation of regularity of the solution for the case $p \neq 2$, a decrease with respect to convergence rates of the method is stated, in general, by retaining smoothly first 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 second step.