Finite element approximation of singular power-law systems

Author:
Adrian Hirn

Journal:
Math. Comp. **82** (2013), 1247-1268

MSC (2010):
Primary 76A05, 35Q35, 65N30, 65N12, 65N15

DOI:
https://doi.org/10.1090/S0025-5718-2013-02668-3

Published electronically:
January 18, 2013

MathSciNet review:
3042563

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Abstract: Non-Newtonian fluid motions are often modeled by a power-law ansatz. In the present paper, we consider shear thinning singular power-law models which feature an unbounded viscosity in the limit of zero shear rate, and we study the finite element (FE) discretization of the equations of motion. In the case under consideration, numerical instabilities usually arise when the FE equations are solved via Newton's method. In this paper, we propose a numerical method that enables the stable approximation of singular power-law systems and that is based on a simple regularization of the power-law model. Our proposed method generates a sequence of discrete functions that is computable in practice via Newton's method and that converges to the exact solution of the power-law system for diminishing mesh size. First, for the regularized model we discuss Newton's method and we show its stability in the sense that we derive an upper bound for the condition number of the Newton matrix. Then, we prove a priori error estimates that quantify the convergence of the proposed method. Finally, we illustrate numerically that our regularized approximation method surpasses the nonregularized one regarding accuracy and numerical efficiency.

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Additional Information

**Adrian Hirn**

Affiliation:
Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany

Email:
adrian.hirn@iwr.uni-heidelberg.de

DOI:
https://doi.org/10.1090/S0025-5718-2013-02668-3

Keywords:
Power-law fluid,
finite element method,
error analysis

Received by editor(s):
March 2, 2011

Received by editor(s) in revised form:
November 4, 2011

Published electronically:
January 18, 2013

Additional Notes:
This work was supported by the International Graduate College IGK 710 “Complex Processes: Modeling, Simulation and Optimization” and the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp) at the Interdisciplinary Center for Scientific Computing (IWR) of the University of Heidelberg

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.