Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year


IMPACT FACTOR 2016: 1.097

CiteScore 2016: 1.09

SCImago Journal Rank (SJR) 2016: 0.872
Source Normalized Impact per Paper (SNIP) 2016: 0.887

Mathematical Citation Quotient (MCQ) 2016: 0.75

Online
ISSN
1609-9389
See all formats and pricing
More options …
Volume 18, Issue 2

Issues

A Study on the Galerkin Least-Squares Method for the Oldroyd-B Model

Tsu-Fen Chen / Hyesuk Lee / Chia-Chen Liu
Published Online: 2017-07-06 | DOI: https://doi.org/10.1515/cmam-2017-0022

Abstract

We consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

Keywords: Viscoelastic Flow; Galerkin Least-Squares

MSC 2010: 65N30

References

  • [1]

    F. P. T. Baaijens, Application of low-order discontinuous Galerkin methods to the analysis of viscoelastic flows, J. Non-Newtonian Fluid Mech. 52 (1994), 37–57. CrossrefGoogle Scholar

  • [2]

    F. P. T. Baaijens, Mixed finite element methods for viscoelastic flow analysis: A review, J. Non-Newtonian Fluid Mech. 79 (1998), 361–385. CrossrefGoogle Scholar

  • [3]

    J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. I: Discontinuous constraints, Numer. Math. 63 (1992), no. 1, 13–27. CrossrefGoogle Scholar

  • [4]

    M. Behr, D. Arora and M. Pasquali, Stabilized finite element methods of GLS type for Maxwell-B and Oldroyd-B viscoelastic fluids, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), http://www.mit.jyu.fi/eccomas2004/proceedings/pdf/728.pdf.

  • [5]

    R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics, John Wiley and Sons, New York, 1987. Google Scholar

  • [6]

    J. Bonvin, M. Picasso and R. Stenberg, GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 29-30, 3893–3914. CrossrefGoogle Scholar

  • [7]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer, New York, 1994. Google Scholar

  • [8]

    T. F. Chen, C. L. Cox, H. C. Lee and K. L. Tung, Least-squares finite element methods for generalized Newtonian and viscoelastic flows, Appl. Numer. Math. 60 (2010), no. 10, 1024–1040. Web of ScienceCrossrefGoogle Scholar

  • [9]

    T.-F. Chen, H. Lee and C.-C. Liu, Numerical approximation of the Oldroyd-B model by the weighted least-squares/discontinuous Galerkin method, Numer. Methods Partial Differential Equations 29 (2013), no. 2, 531–548. Web of ScienceCrossrefGoogle Scholar

  • [10]

    O. M. Coronado, D. Arora, M. Behr and M. Pasquali, Four-field Galerkin/least-squares formulation for viscoelastic fluids, J. Non-Newtonian Fluid Mech. 140 (2006), 132–144. CrossrefGoogle Scholar

  • [11]

    V. J. Ervin, H. Lee and L. N. Ntasin, Analysis of the Oseen-viscoelastic fluid flow problem, J. Non-Newtonian Fluid Mech. 127 (2005), 157–168. CrossrefGoogle Scholar

  • [12]

    A. Fortin, R. Guénette and R. Pierre, On the discrete EVSS method, Comput. Methods Appl. Mech. Engrg. 189 (2000), no. 1, 121–139. CrossrefGoogle Scholar

  • [13]

    V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. Google Scholar

  • [14]

    F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. Web of ScienceGoogle Scholar

  • [15]

    E. Jenkins and H. Lee, A domain decomposition method for the Oseen-viscoelastic flow equations, Appl. Math. Comput. 195 (2008), no. 1, 127–141. Web of ScienceGoogle Scholar

  • [16]

    J. M. Kim, C. Kim, J. H. Kim, C. Chunga, K. H. Ahna and S. J. Lee, High-resolution finite element simulation of 4:1 planar contraction flow of viscoelastic fluid, J. Non-Newtonian Fluid Mech. 129 (2005), 23–37. CrossrefGoogle Scholar

  • [17]

    S. D. Kim and B. C. Shin, H-1 least-squares method for the velocity-pressure-stress formulation of Stokes equations, Appl. Numer. Math. 40 (2002), no. 4, 451–465. Google Scholar

  • [18]

    H. Lee, A multigrid method for viscoelastic fluid flow, SIAM J. Numer. Anal. 42 (2004), no. 1, 109–129. CrossrefGoogle Scholar

  • [19]

    H.-C. Lee and T.-F. Chen, A nonlinear weighted least-squares finite element method for Stokes equations, Comput. Math. Appl. 59 (2010), no. 1, 215–224. CrossrefWeb of ScienceGoogle Scholar

  • [20]

    A. Liakos and H. Lee, Two-level finite element discretization of viscoelastic fluid flow, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 44-46, 4965–4979. CrossrefGoogle Scholar

  • [21]

    M. A. Mendelson, P. W. Yeh, R. C. Armstrong and R. A. Brown, Approximation error in finite element calculation of viscoelastic fluid flows, J. Non-Newtonian Fluid Mech. 10 (1982), 31–54. CrossrefGoogle Scholar

  • [22]

    K. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Numer. Math. 72 (1995), no. 2, 223–238. CrossrefGoogle Scholar

  • [23]

    M. Picasso and J. Rappaz, Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows, M2AN Math. Model. Numer. Anal. 35 (2001), no. 5, 879–897. CrossrefGoogle Scholar

  • [24]

    D. Rajagopalan, R. C. Armstrong and R. A. Brown, Finite element methods for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity, J. Non-Newtonian Fluid Mech. 36 (1990), 159–192. CrossrefGoogle Scholar

  • [25]

    M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech. 65 (1985), no. 9, 449–451. CrossrefGoogle Scholar

  • [26]

    D. Sandri, Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. Continuous approximation of the stress, SIAM J. Numer. Anal. 31 (1994), no. 2, 362–377. CrossrefGoogle Scholar

  • [27]

    O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I: The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. CrossrefGoogle Scholar

About the article

Received: 2017-01-24

Revised: 2017-06-05

Accepted: 2017-06-19

Published Online: 2017-07-06

Published in Print: 2018-04-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1418960

The first author was partially supported by the National Science Council of Taiwan under contract no. 101-2115-M-194-009. The second author was partially supported by the NSF under grant no. DMS-1418960.


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 2, Pages 181–198, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0022.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in