Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

TL;DR

This paper introduces a novel mixed finite element method for the Oseen equations with variable viscosity, incorporating velocity, vorticity, and pressure, and provides theoretical analysis and numerical validation.

Contribution

It develops an augmented mixed finite element scheme for the Oseen problem with variable viscosity, including stability analysis and error estimates, applicable to various finite element pairs.

Findings

The method is stable and convergent under standard finite element assumptions.

Numerical tests confirm theoretical convergence rates.

Adaptive algorithms improve solution accuracy efficiently.

Abstract

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.