Finite element discretization of the steady, generalized Navier-Stokes equations with inhomogeneous Dirichlet boundary conditions

TL;DR

This paper introduces a finite element method for steady generalized Navier-Stokes equations with shear-dependent viscosity, inhomogeneous boundary conditions, and divergence constraints, providing convergence proofs and error estimates.

Contribution

It presents a novel finite element discretization for complex Navier-Stokes models with theoretical convergence and error analysis.

Findings

Discrete solutions converge weakly to the true solution.

A priori error estimates are established for velocity and pressure.

Numerical experiments validate theoretical results.

Abstract

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.