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Semi-implicit Euler Scheme for Generalized Newtonian Fluids

Rheological behavior of certain non-Newtonian fluids in engineering sciences is often modeled by power law ansatzes with p \leq 2. So far, existing numerical analysis for local strong solutions studies a fully implicit time discretization and find only restricted ranges of admissible p's for corresponding error estimates [A. Prohl and M. Ruzicka, SIAM J. Numer. Anal., 39 (2001), pp. 214-249]; different nonlinear stabilization strategies which allow a corresponding error analysis for smaller p's are examined in [L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, Ph. D. thesis, University of Freiburg, Freiburg, Germany, 2002] and [L. Diening, A. Prohl, and M. Ruzicka, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, New York, 2002, pp. 89-118]. In the present paper, a semi-implicit time discretization scheme is proposed, and error estimates apply to the extended range p \in (3/2 ,2]. The key analytical tool is a new Gronwall-type inequality.