Optimal Error Estimates for a Semi-Implicit Euler Scheme for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behavior of fluids in engineering sciences is modeled by power law ansatz with p ∈ (1, 2]. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal O(k2) error estimate, where k is the time step size. This improves results in [L. Diening, A. Prohl, and M. Růžička, SIAM J. Numer. Anal., 44 (2006), pp. 1172–1190], which where suboptimal in terms of the order of convergence. Our results hold in three-dimensional domains (with periodic boundary conditions) for the range p ∈ (3/2, 2] and are uniform with respect to the degeneracy parameter δ ∈ [0, δ0] of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved.