Non-conforming finite element discretisations of a class of non-Newtonian flows

We study an approximation for a class of non-Newtonian model, based on the general Ladyzhenskaya law for continuous and non-linear kinematic viscosity functions of the fluid. We introduce a non-conforming finite element approximation of non-linear Stokes problem; we give existence, uniqueness results and optimal error estimates under minimal regularity assumptions. Finally, we introduce a post-processing to obtain conforming approximation of stress, which allow to derive various equivalence with two or three field mixed dual approximations of non-linear Stokes problem.