We develop the a posteriori error analysis of finite element approximations to implicit power-law-like models for viscous incompressible fluids in space dimensions, . 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 -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].
Accepted:
DOI: 10.1051/m2an/2015085
Keywords: Adaptive finite element methods, implicit constitutive models, power-law fluids, a posteriori analysis, convergence
@article{M2AN_2016__50_5_1333_0, author = {Kreuzer, Christian and S\"uli, Endre}, title = {Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1333--1369}, publisher = {EDP-Sciences}, volume = {50}, number = {5}, year = {2016}, doi = {10.1051/m2an/2015085}, zbl = {1457.65201}, mrnumber = {3554545}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015085/} }
TY - JOUR AU - Kreuzer, Christian AU - Süli, Endre TI - Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1333 EP - 1369 VL - 50 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015085/ DO - 10.1051/m2an/2015085 LA - en ID - M2AN_2016__50_5_1333_0 ER -
%0 Journal Article %A Kreuzer, Christian %A Süli, Endre %T Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1333-1369 %V 50 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015085/ %R 10.1051/m2an/2015085 %G en %F M2AN_2016__50_5_1333_0
Kreuzer, Christian; Süli, Endre. Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1333-1369. doi : 10.1051/m2an/2015085. http://www.numdam.org/articles/10.1051/m2an/2015085/
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