Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

TL;DR

This paper analyzes finite element methods for simulating steady incompressible fluid flows with complex, implicit power-law rheology, proving convergence of solutions using advanced mathematical techniques.

Contribution

It introduces a novel finite element analysis framework for implicit power-law models, including a new Lipschitz truncation technique for Sobolev functions.

Findings

Finite element solutions converge to weak solutions as mesh size decreases.

The analysis handles multi-valued maximal monotone r-graphs in stress-strain relations.

A new Lipschitz truncation method for Sobolev functions is developed.

Abstract

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi--valued, maximal monotone $r$-graph, with $1<r<\infty$. Using a variety of weak compactness techniques, including Chacon's biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter $h$ tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions.