# Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

## TL;DR

This paper presents a novel mixed finite element formulation for Brinkman equations with variable viscosity, providing stability analysis and optimal error estimates, validated by computational experiments.

## Contribution

It introduces a non-symmetric mixed finite element approach for Brinkman equations with variable viscosity, extending stability and error analysis to arbitrary order vorticity discretizations.

## Key findings

- Stable finite element coupling for velocity, pressure, and vorticity.
- Optimal a priori error estimates established.
- Computational examples confirm theoretical results.

## Abstract

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity and pressure with non-constant viscosity. The analysis is performed by the classical Babu\v{s}ka-Brezzi theory, and we state that any inf-sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates which are further confirmed through computational examples