Finite element approximation of singular powerlaw systems
Author:
Adrian Hirn
Journal:
Math. Comp. 82 (2013), 12471268
MSC (2010):
Primary 76A05, 35Q35, 65N30, 65N12, 65N15
Published electronically:
January 18, 2013
MathSciNet review:
3042563
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Additional Information
Abstract: NonNewtonian fluid motions are often modeled by a powerlaw ansatz. In the present paper, we consider shear thinning singular powerlaw 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 powerlaw systems and that is based on a simple regularization of the powerlaw 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 powerlaw 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.
 1.
Chérif
Amrouche and Vivette
Girault, Decomposition of vector spaces and application to the
Stokes problem in arbitrary dimension, Czechoslovak Math. J.
44(119) (1994), no. 1, 109–140. MR 1257940
(95c:35190)
 2.
John
W. Barrett and W.
B. Liu, Quasinorm error bounds for the finite element
approximation of a nonNewtonian flow, Numer. Math.
68 (1994), no. 4, 437–456. MR 1301740
(95h:65078), 10.1007/s002110050071
 3.
R.
Becker and M.
Braack, A finite element pressure gradient stabilization for the
Stokes equations based on local projections, Calcolo
38 (2001), no. 4, 173–199. MR 1890352
(2002m:65112), 10.1007/s1009200181804
 4.
L.
Belenki, L.
C. Berselli, L.
Diening, and M.
R\ocirc{u}žička, On the finite element approximation
of 𝑝Stokes systems, SIAM J. Numer. Anal. 50
(2012), no. 2, 373–397. MR
2914267, 10.1137/10080436X
 5.
Luigi
C. Berselli, Lars
Diening, and Michael
R\ocirc{u}žička, Existence of strong solutions for
incompressible fluids with shear dependent viscosities, J. Math. Fluid
Mech. 12 (2010), no. 1, 101–132. MR 2602916
(2011c:76005), 10.1007/s000210080277y
 6.
D. V. Boger, A. Cabelli and A. L. Halmos, The behavior of a powerlaw fluid flowing through a sudden expansion, AIChE Journal 21 (1975), no. 3, 540549.
 7.
M.
Braack, E.
Burman, V.
John, and G.
Lube, Stabilized finite element methods for the generalized Oseen
problem, Comput. Methods Appl. Mech. Engrg. 196
(2007), no. 46, 853–866. MR 2278180
(2007i:76065), 10.1016/j.cma.2006.07.011
 8.
Susanne
C. Brenner and L.
Ridgway Scott, The mathematical theory of finite element
methods, Texts in Applied Mathematics, vol. 15, SpringerVerlag,
New York, 1994. MR 1278258
(95f:65001)
 9.
P. G. Ciarlet, The finite elements methods for elliptic problems, NorthHolland, 1980.
 10.
Peter
Deuflhard, Newton methods for nonlinear problems, Springer
Series in Computational Mathematics, vol. 35, SpringerVerlag, Berlin,
2004. Affine invariance and adaptive algorithms. MR 2063044
(2005h:65002)
 11.
Lars
Diening and Frank
Ettwein, Fractional estimates for nondifferentiable elliptic
systems with general growth, Forum Math. 20 (2008),
no. 3, 523–556. MR 2418205
(2009h:35101), 10.1515/FORUM.2008.027
 12.
Giovanni
P. Galdi, Rolf
Rannacher, Anne
M. Robertson, and Stefan
Turek, Hemodynamical flows, Oberwolfach Seminars,
vol. 37, Birkhäuser Verlag, Basel, 2008. Modeling, analysis and
simulation; Lectures from the seminar held in Oberwolfach, November
20–26, 2005. MR 2416195
(2009e:76003)
 13.
GASCOIGNE, The finite element toolkit, http://www.gascoigne.unihd.de.
 14.
John
G. Heywood, Rolf
Rannacher, and Stefan
Turek, Artificial boundaries and flux and pressure conditions for
the incompressible NavierStokes equations, Internat. J. Numer.
Methods Fluids 22 (1996), no. 5, 325–352. MR 1380844
(97f:76045), 10.1002/(SICI)10970363(19960315)22:5<325::AIDFLD307>3.0.CO;2Y
 15.
A. Hirn, Approximation of the Stokes equations with equalorder finite elements, J. Math. Fluid Mech. (2012), doi:10.1007/s0002101200950.
 16.
A. Hirn, M. Lanzendörfer and J. Stebel, Finite element approximation of flow of fluids with shear rate and pressure dependent viscosity, IMA Journal of Numerical Analysis (2012), doi: 10.1093/imanum/drr033.
 17.
Martin
Lanzendörfer and Jan
Stebel, On pressure boundary conditions for steady flows of
incompressible fluids with pressure and shear rate dependent
viscosities, Appl. Math. 56 (2011), no. 3,
265–285. MR 2800578
(2012g:35260), 10.1007/s1049201100161
 18.
J.
Málek, J.
Nečas, M.
Rokyta, and M.
R\ocirc{u}žička, Weak and measurevalued solutions to
evolutionary PDEs, Applied Mathematics and Mathematical Computation,
vol. 13, Chapman & Hall, London, 1996. MR 1409366
(97g:35002)
 19.
J.
Málek and K.
R. Rajagopal, Mathematical issues concerning the NavierStokes
equations and some of its generalizations, Evolutionary equations.
Vol. II, Handb. Differ. Equ., Elsevier/NorthHolland, Amsterdam, 2005,
pp. 371–459. MR 2182831
(2006k:35221)
 20.
J.
Málek, K.
R. Rajagopal, and M.
R\ocirc{u}žička, Existence and regularity of
solutions and the stability of the rest state for fluids with shear
dependent viscosity, Math. Models Methods Appl. Sci.
5 (1995), no. 6, 789–812. MR 1348587
(96i:76002), 10.1142/S0218202595000449
 21.
L. J. Sonder and P. C. England, Vertical averages of rheology of the continental lithosphere, Earth Planet. Sci. Lett. 77 (1986), 8190.
 1.
 C. Amrouche and V. Girault, Decomposition of vectorspaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44 (1994), no. 1, 109140. MR 1257940 (95c:35190)
 2.
 J. W. Barrett and W. B. Liu, Quasinorm error bounds for the finite element approximation of a nonNewtonian flow, Numer. Math. 68 (1994), 437456. MR 1301740 (95h:65078)
 3.
 R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (2001), 173199. MR 1890352 (2002m:65112)
 4.
 L. Belenki, L. C. Berselli, L. Diening and M. Růžička, On the finite element approximation of Stokes systems, SIAM J. Numer. Anal. 50 (2012), no. 2, 373397. MR 2914267
 5.
 L. C. Berselli, L. Diening and M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech. 12 (2010), 101132. MR 2602916 (2011c:76005)
 6.
 D. V. Boger, A. Cabelli and A. L. Halmos, The behavior of a powerlaw fluid flowing through a sudden expansion, AIChE Journal 21 (1975), no. 3, 540549.
 7.
 M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem, Comput. Methods Appl. Mech. Engrg. 196 (2007), 853866. MR 2278180 (2007i:76065)
 8.
 S. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods, SpringerVerlag, New York, 1994. MR 1278258 (95f:65001)
 9.
 P. G. Ciarlet, The finite elements methods for elliptic problems, NorthHolland, 1980.
 10.
 P. Deuflhard, Newtonmethods for nonlinear problems  affine invariance and adaptive algorithms, SpringerVerlag, Berlin Heidelberg, 2004. MR 2063044 (2005h:65002)
 11.
 L. Diening and F. Ettwein, Fractional estimates for nondifferentiable elliptic systems with general growth, Forum Math. 20 (2008), 523556. MR 2418205 (2009h:35101)
 12.
 G. P. Galdi, R. Rannacher, A. M. Robertson and S. Turek, Hemodynamical Flows  Modeling, Analysis and Simulation, Oberwolfach Seminars, vol. 37, Birkhäuser, Basel, 2008. MR 2416195 (2009e:76003)
 13.
 GASCOIGNE, The finite element toolkit, http://www.gascoigne.unihd.de.
 14.
 J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible NavierStokes equations, Int. J. Num. Meth. Fluids 22 (1996), 325352. MR 1380844 (97f:76045)
 15.
 A. Hirn, Approximation of the Stokes equations with equalorder finite elements, J. Math. Fluid Mech. (2012), doi:10.1007/s0002101200950.
 16.
 A. Hirn, M. Lanzendörfer and J. Stebel, Finite element approximation of flow of fluids with shear rate and pressure dependent viscosity, IMA Journal of Numerical Analysis (2012), doi: 10.1093/imanum/drr033.
 17.
 M. Lanzendörfer and J. Stebel, On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities, Appl. Math. 56 (2011), no. 3, 265285. MR 2800578
 18.
 J. Málek, J. Nečas, M. Rokyta, and M. Růžička, Weak and measurevalued solutions to evolutionary PDEs, Chapman & Hall, London, 1996. MR 1409366 (97g:35002)
 19.
 J. Málek and K. R. Rajagopal, Handbook of Differential Equations: Evolutionary equations, vol. 2, Chapter 5, Mathematical issues concerning the NavierStokes equations and some of its generalizations, pp. 371459, Elsevier/NorthHolland, Amsterdam, 2005. MR 2182831 (2006k:35221)
 20.
 J. Málek, K. R. Rajagopal and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci. 5 (1995), 789812. MR 1348587 (96i:76002)
 21.
 L. J. Sonder and P. C. England, Vertical averages of rheology of the continental lithosphere, Earth Planet. Sci. Lett. 77 (1986), 8190.
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Additional Information
Adrian Hirn
Affiliation:
Institut für Angewandte Mathematik, RuprechtKarlsUniversität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany
Email:
adrian.hirn@iwr.uniheidelberg.de
DOI:
http://dx.doi.org/10.1090/S002557182013026683
PII:
S 00255718(2013)026683
Keywords:
Powerlaw 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.
