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

¹ Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44801 Bochum, Germany.
christan.kreuzer@rub.de
² Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK.
Endre.Suli@maths.ox.ac.uk

Received: 20 March 2015
Revised: 16 September 2015
Accepted: 30 October 2015

Abstract

We develop the a posteriori error analysis of finite element approximations to implicit power-law-like models for viscous incompressible fluids in d space dimensions, d ∈ {2,3}. 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 . 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. Süli, SIAM J. Numer. Anal. 51 (2013) 984–1015].