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 $p>\frac{2d}{d+2}$. The Dirichlet boundary condition is imposed strongly, using any discretization of the boundary data which converges at a sufficient rate. $\mathit{\text{A}}\mathit{\text{priori}}$ error estimates for the velocity vector field and kinematic pressure are derived and numerical experiments are conducted. These confirm the quasi-optimality of the $\mathit{\text{a}}\mathit{\text{priori}}$ error estimate for the velocity vector field. The $\mathit{\text{a}}\mathit{\text{priori}}$ error estimates for the kinematic pressure are quasi-optimal if $p\le 2$.