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In this paper we study the numerical error arising in the space-timeapproximation of unsteady generalized Newtonian fluids which possess astress-tensor with $(p,\delta)$-structure. A semi-implicit time-discretizationscheme coupled with conforming inf-sup stable finite element spacediscretization is analyzed. The main result, which improves previous suboptimalestimates as those in [A. Prohl, and M. Ruzicka, SIAM J. Numer. Anal., 39(2001), pp. 214--249] is the optimal $O(k+h)$ error-estimate valid in the range$p\in (3/2,2]$, where $k$ and $h$ are the time-step and the mesh-size,respectively. Our results hold in three-dimensional domains (with periodicboundary conditions) and are uniform with respect to the degeneracy parameter$\in [0,\delta_0]$ of the extra stress tensor.

Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids