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

In this work, we propose and analyse an augmented mixed finite element method for solving the Navier-Stokes equations describing the motion of incompressible fluid. The model is written in terms of velocity, vorticity, and pressure, and takes into account non-constant viscosity and no-slip boundary conditions. The weak formulation of the method includes least-squares terms that arise from the constitutive equation and the incompressibility condition. We discuss the theoretical and practical implications of using augmentation in detail. Additionally, 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 and provide a set of numerical tests in 2D and 3D to illustrate the behaviour of the discretisations and verify their theoretical convergence rates. Overall, this method provides an efficient and accurate solution for simulating fluid flow in a wide range of scenarios.