On augmented finite element formulation for the Navier--Stokes equations with vorticity and variable viscosity

TL;DR

This paper introduces an augmented mixed finite element method for the Navier--Stokes equations incorporating vorticity and variable viscosity, with theoretical analysis and numerical validation of convergence and stability.

Contribution

It develops a novel augmented formulation for Navier--Stokes that handles vorticity and variable viscosity, providing optimal error estimates and flexible finite element choices.

Findings

Proves existence and uniqueness of solutions under small data conditions.

Establishes optimal a priori error estimates for the scheme.

Numerical tests confirm theoretical convergence rates in 2D and 3D.

Abstract

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use fixed--point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.