A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis

Abstract

We present a mixed finite element method for a class of non-linear Stokes models arising in quasi-Newtonian fluids. Our results include, as a by-product, a new mixed scheme for the linear Stokes equation. The approach is based on the introduction of both the flux and the tensor gradient of the velocity as further unknowns, which yields a twofold saddle point operator equation as the resulting variational formulation. We prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. The corresponding Galerkin scheme is defined by using piecewise constant functions and Raviart–Thomas spaces of lowest order.