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

TL;DR

This paper introduces a finite element discretization method for the steady generalized Navier-Stokes equations applicable to all shear stress exponents above a certain threshold, with proven error estimates and numerical validation.

Contribution

It develops a divergence reconstruction-based FE formulation valid for all shear exponents p > 2d/(d+2), with strong boundary condition enforcement and error analysis.

Findings

Error estimates for velocity are quasi-optimal.

Error estimates for pressure are quasi-optimal when p ≤ 2.

Numerical experiments confirm theoretical error bounds.

Abstract

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 $p \leq 2$.