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

In this paper we study the numerical error arising in the space-time approximation of unsteady generalized Newtonian fluids which possess a stress-tensor with $(p,\delta)$-structure. A semi-implicit time-discretization scheme coupled with conforming inf-sup stable finite element space discretization is analyzed. The main result, which improves previous suboptimal estimates 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 periodic boundary conditions) and are uniform with respect to the degeneracy parameter $\in [0,\delta_0]$ of the extra stress tensor.