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

TL;DR

This paper develops an a posteriori error analysis for finite element methods applied to steady incompressible fluid flows with implicit power-law-like rheology, establishing bounds, stability, and convergence of adaptive algorithms.

Contribution

It introduces a novel a posteriori error analysis framework for implicit power-law models, including stability and convergence results for adaptive finite element approximations.

Findings

Established upper and lower bounds on finite element residuals.

Proved weak convergence of adaptive algorithms to weak solutions.

Applied advanced compactness techniques like Chacon's biting lemma.

Abstract

We develop the a posteriori error 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 $\frac{2d}{d+1}<r<\infty$. We establish upper and lower bounds on the finite element residual, as well as the local stability of the error bound. We then consider an adaptive finite element approximation of the problem, and, under suitable assumptions, we show the weak convergence of the adaptive algorithm to a weak solution of the boundary-value problem. The argument is based on a variety of weak compactness techniques, including Chacon's biting lemma and a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions, introduced by L. Diening, C. Kreuzer and E. S\"uli [Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal., 51(2), 984--1015].