Finite element discretization of the steady, generalized Navier-Stokes equations for small shear stress exponents
This work provides a theoretical and numerical framework for a class of non-Newtonian fluid models, but the results are incremental as they extend existing techniques to a wider range of exponents.
The paper proposes a finite element discretization for the steady, generalized Navier-Stokes equations valid for shear stress exponents p > 2d/(d+2), and derives a priori error estimates for velocity and pressure, with numerical experiments confirming quasi-optimality for velocity and for pressure when p ≤ 2.
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 $p \leq 2$.