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

TL;DR

The work addresses the finite element approximation of steady generalized Navier–Stokes equations with $(p,\delta)$-structure on Lipschitz domains, valid for all $p>\frac{2d}{d+2}$. It introduces divergence-free reconstruction operators to enable a full $p$-range analysis and allows flexible discrete Dirichlet boundary data, including nodal interpolation. The authors derive a priori error estimates for velocity and pressure, proving quasi-optimal convergence for the velocity for $p\in(\frac{2d}{d+2},\frac{2d}{d+1})$ and quasi-optimal pressure estimates for $p\le 2$, supported by a robust numerical study on the unit square. The numerical experiments confirm the theoretical rates and demonstrate the practical viability of the divergence-reconstruction approach in handling inhomogeneous Dirichlet data and non-Newtonian behavior in a FE framework.

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 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 $p \leq 2$.

Theorems & Definitions (43)

• Proposition 2.2
• Proof 1
• Lemma 2.4
• Proof 2
• Definition 3.2: finite element spaces
• Remark 3.6
• Lemma 3.7: discrete inf-sup condition
• Proof 3
• Lemma 3.10: classical interpolation operators
• Proof 4