Finite element discretization of the steady, generalized Navier-Stokes equations for small shear stress exponents

A finite element (FE) discretization for the steady, incompressible, fully inhomogeneous, generalized Navier-Stokes equations is proposed. By the method of divergence reconstruction operators, the formulation is valid for all shear stress exponents $p > \tfrac{2d}{d+2}$. The Dirichlet boundary condition is imposed strongly, using any discretization of the boundary data which converges at a sufficient rate. $\textit{A priori}$ error estimates for the velocity vector field and kinematic pressure are derived and numerical experiments are conducted. These confirm the quasi-optimality of the $\textit{a priori}$ error estimate for the velocity vector field. The $\textit{a priori}$ error estimates for the kinematic pressure are quasi-optimal if $p \leq 2$.