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 multivalued, 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.

Keywords

  1. finite element methods
  2. implicit constitutive models
  3. power-law fluids
  4. convergence
  5. weak compactness
  6. discrete Lipschitz truncation

MSC codes

  1. 65N30
  2. 65N12
  3. 76A05
  4. 35Q35

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Information & Authors

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 984 - 1015
ISSN (online): 1095-7170

History

Submitted: 10 April 2012
Accepted: 26 December 2012
Published online: 21 March 2013

Keywords

  1. finite element methods
  2. implicit constitutive models
  3. power-law fluids
  4. convergence
  5. weak compactness
  6. discrete Lipschitz truncation

MSC codes

  1. 65N30
  2. 65N12
  3. 76A05
  4. 35Q35

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