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 space-time approximation of unsteady power-law non-Newtonian fluids. A semi-implicit time discretization scheme, coupled with space discretization made with conforming finite elements 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 wide range pin]3/2,2], where k is the time-step and h the mesh-size. Our results hold in three-dimensional domains (with periodic boundary conditions), are uniform with respect to the degeneracy parameter delta of the extra stress tensor, and a stability h-k-coupling depending on p is also needed.