`Discrete Dynamics in Nature and SocietyVolume 2011, Article ID 689804, 25 pageshttp://dx.doi.org/10.1155/2011/689804`
Research Article

## A Defect-Correction Method for Time-Dependent Viscoelastic Fluid Flow Based on SUPG Formulation

1School of Science, Xi'an Jiaotong University, Xi'an 710049, China
2School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China
3Institute of Mathematics Science, Yunnan Normal University, Kunming 650092, China
4Department of Mathematics, Yunnan Nationalities University, Kunming 650031, China

Received 22 November 2010; Revised 13 March 2011; Accepted 29 April 2011

Academic Editor: M. De la Sen

Copyright © 2011 Yunzhang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

1. J. Baranger and S. Wardi, “Numerical analysis of a FEM for a transient viscoelastic flow,” Computer Methods in Applied Mechanics and Engineering, vol. 125, no. 1–4, pp. 171–185, 1995.
2. V. J. Ervin and W. W. Miles, “Approximation of time-dependent viscoelastic fluid flow: SUPG approximation,” SIAM Journal on Numerical Analysis, vol. 41, no. 2, pp. 457–486, 2003.
3. V. J. Ervin and W. W. Miles, “Approximation of time-dependent, multi-component, viscoelastic fluid flow,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 18-20, pp. 2229–2255, 2005.
4. T. J. Hughes and A. N. Brooks, A multidimensional Upwind Scheme with No Crosswind Diffusion, in Finite Element Methods for Convection Dominated Flows, New York, NY, USA, 1979.
5. M. Bensaada and D. Esselaoui, “Error estimates for a stabilized finite element method for the Oldroyd B model,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1042–1059, 2007.
6. V. J. Ervin and N. Heuer, “Approximation of time-dependent, viscoelastic fluid flow: Crank-Nicolson, finite element approximation,” Numerical Methods for Partial Differential Equations, vol. 20, no. 2, pp. 248–283, 2004.
7. J. C. Chrispell, V. J. Ervin, and E. W. Jenkins, “A fractional step $\theta -$method approximation of time-dependent viscoelastic fluid flow,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 159–175, 2009.
8. A. Bonito, P. Clément, and M. Picasso, “Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow,” Numerische Mathematik, vol. 107, no. 2, pp. 213–255, 2007.
9. P. Nithiarasu, “A fully explicit characteristic based split (CBS) scheme for viscoelastic flow calculations,” International Journal for Numerical Methods in Engineering, vol. 60, no. 5, pp. 949–978, 2004.
10. H. K. Lee, “Analysis of a defect correction method for viscoelastic fluid flow,” Computers and Mathematics with Applications, vol. 48, no. 7-8, pp. 1213–1229, 2004.
11. V. J. Ervin and H. Lee, “Defect correction method for viscoelastic fluid flows at high Weissenberg number,” Numerical Methods for Partial Differential Equations, vol. 22, no. 1, pp. 145–164, 2006.
12. V. J. Ervin, J. S. Howell, and H. Lee, “A two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 818–834, 2008.
13. X. L. Luo, “An incremental difference formulation for viscoelastic flows and high resolution FEM solutions at high Weissenberg numbers,” Journal of Non-Newtonian Fluid Mechanics, vol. 79, no. 1, pp. 57–75, 1998.
14. J. Petera, “A new finite element scheme using the Lagrangian framework for simulation of viscoelastic fluid flows,” Journal of Non-Newtonian Fluid Mechanics, vol. 103, no. 1, pp. 1–43, 2002.
15. J. S. Howell, “Computation of viscoelastic fluid flows using continuation methods,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 187–201, 2009.
16. K. Bhmer and H. J. Stetter, Defect Correction Methods-Theory and Applications, vol. 5 of Computing Supplementum, Springer, Vienna, Austria, 1984.
17. R. Minero, M. J. H. Anthonissen, and R. M. M. Mattheij, “A local defect correction technique for time-dependent problems,” Numerical Methods for Partial Differential Equations, vol. 22, no. 1, pp. 128–144, 2006.
18. V. J. Ervin, W. J. Layton, and J. M. Maubach, “Adaptive defect-correction methods for viscous incompressible flow problems,” SIAM Journal on Numerical Analysis, vol. 37, no. 4, pp. 1165–1185, 2000.
19. W. Layton, H. Lee, and J. A. Peterson, “A defect-correction method for the incompressible Navier-Stokes equations,” Applied Mathematics and Computation, vol. 129, no. 1, pp. 1–19, 2002.
20. Q. Liu and Y. R. Hou, “A two-level defect-correction method for Navier-Stokes equations,” Bulletin of the Australian Mathematical Society, vol. 81, no. 3, pp. 442–454, 2010.
21. A. Labovschii, “A defect correction method for the time-dependent Navier-Stokes equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 1, pp. 1–25, 2009.
22. Y. Zhang, Y. Hou, and B. Mu, “Defect correction method for time-dependent viscoelastic fluid flow,” International Journal of Computer Mathematics, vol. 88, no. 7, pp. 1546–1563, 2011.
23. R. B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, John Wiley and Sons, New York, NY, USA, 1987.
24. C. Guillopé and J. C. Saut, “Existence results for the flow of viscoelastic fluids with a differential constitutive law,” Nonlinear Analysis. Theory, Methods & Applications, vol. 15, no. 9, pp. 849–869, 1990.
25. M. Renardy, Mathematical Analysis of Viscoelastic Flows, vol. 73 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2000.
26. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1994.
27. V. Girault and P. A. Raviart, Finite Element methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 1986.
28. P. Clément, “Approximation by finite element functions using local regularization,” RAIRO Analyse Numerique, vol. 9, no. R-2, pp. 77–84, 1975.
29. J. G. Heywood and R. Rannacher, “Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization,” SIAM Journal on Numerical Analysis, vol. 27, no. 2, pp. 353–384, 1990.
30. F. Hecht, O. Pironneau, A. Le Hyaric, and K. Ohtsuka, “FreeFem++,” 2009, http://www.freefem.org/ff++.
31. H. Lee, “A multigrid method for viscoelastic fluid flow,” SIAM Journal on Numerical Analysis, vol. 42, no. 1, pp. 109–129, 2004.
32. F. P.T. Baaijens, “Mixed finite element methods for viscoelastic flow analysis: a review,” Journal of Non-Newtonian Fluid Mechanics, vol. 79, no. 2-3, pp. 361–385, 1998.
33. R. G. Owens and T. N. Phillips, Computational Rheology, Imperial College Press, London, UK, 2002.