Analysis of a stabilised finite element method for power-law fluids

TL;DR

This paper develops and analyzes a stabilized finite element method for simulating incompressible non-Newtonian power-law fluids, ensuring convergence across a broad range of rheological behaviors.

Contribution

It introduces a low-order stabilized finite element scheme with pressure jump stabilization that guarantees convergence for all known existence ranges of solutions.

Findings

Proves convergence of the method for all r > 2d/(d+2)

Constructs a divergence-free velocity approximation

Ensures stability and accuracy for power-law fluid simulations

Abstract

A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise constant approximation of the pressure. Stabilisation, in the form of pressure jumps, is added to the formulation to compensate for the failure of the inf-sup condition, and using an appropriate lifting of the pressure jumps a divergence-free approximation to the velocity field is built and included in the discretisation of the convection term. This construction allows us to prove the convergence of the resulting finite element method for the entire range $r>\frac{2 d}{d+2}$ of the power-law index $r$ for which weak solutions to the model are known to exist in $d$ space dimensions, $d \in \{2,3\}$.