Abstract
In this paper, a new stabilized finite element method based on two local Gauss integrations is considered for the two-dimensional viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newtonian fluid flows. This new stabilized method presents attractive features such as being parameter-free, or being defined for non-edge-based data structures. It confirms that the lowest equal-order P 1 − P 1 triangle element and Q 1 − Q 1 quadrilateral element are compatible. Moreover, the long time stabilities and error estimates for the velocity in H 1-norm and for the pressure in L 2-norm are obtained. Finally, some numerical experiments are performed, which show that the new method is applied to this model successfully and can save lots of computational cost compared with the standard ones.
Similar content being viewed by others
References
Adams, R.: Sobolev Spaces. Academic, New York (1975)
Agranovich, Y., Sobolevskii, P.: Motion of non-linear viscoelastic fluid. Nonlinear Anal. 32, 755–760 (1998)
Akhmatov, M., Oskolkov, A.: On convergence difference schemes for the equations of motion of an Oldroyd fluid. J. Sov. Math. 47, 2926–2933 (1989)
Baaijens, F.: Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newton. Fluid Mech. 79, 361–385 (1998)
Baranger, J., Sandri, D.: Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. I. Discontinuous constraints. Numer. Math. 63, 13–27 (1992)
Bird, R., Armstrong, R., Hassager, O.: Dynamics of Polymeric Liquids, vol. 1. Fluid Mechanics. Wiley, New York (1997)
Bonito, A., Burman, E.: A continuous interior penalty method for viscoelastic flows. SIAM J. Sci. Comput. 30, 1156–1177 (2008)
Cannon, J., Ewing, R., He, Y., Lin, Y.: A modified nonlinear Galerkin method for the viscoelastic fluid motion equations. Int. J. Eng. Sci. 37, 1643–1662 (1999)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Douglas, J., Wang, J.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52, 495–508 (1989)
Ervin, V., Mikes, W.: Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal. 41, 457–486 (2003)
Giraut, V., Raviart, P.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979)
Goswami, D., Pani, A.: A priori error estimates for semidiscrete finite element approximations to equations of motion arising in Oldroyd fluids of order one. Int. J. Numer. Anal. Model. 8, 324–352 (2011)
Guenette, R., Fortin, M.: A new mixed finite method for computing viscoelastic flows. J. Non-Newton. Fluid Mech. 60, 27–52 (1995)
He, Y., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations. Appl. Numer. Math. 58, 1503–1514 (2008)
He, Y., Li, Y.: Asymptotic behavior of linearized viscoelastic flow problem. Discrete Continuous Dyn. Syst. Ser. B 10, 843–856 (2008)
He, Y., Lin, Y., Sun, W.: Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete Continuous Dyn. Syst. Ser. B 6, 41–68 (2006)
He, Y., Lin, Y., Sun, W., Shen, S., Tait, R.: Finite element approximation for the viscoelastic fluid motion problem. J. Comput. Appl. Math. 155, 201–222 (2003)
Heywood, J., Rannacher, R.: Finite-element approximation of the nonstationary Navier-Stokes problem part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)
Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)
Hughes, T., Franca, L.: A new finite element formulation for CFD. VII: the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity, pressure spaces. Comput. Methods Appl. Mech. Eng. 65, 85–97 (1987)
Joseph, D.: Fluid Dynamics of Viscoelastic Liquids. Springer, New York (1990)
Lee, H.: A multigrid method for viscoelastic fluid flow. SIAM J. Numer. Anal. 42, 109–129 (2004)
Li, J., He, Y.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008)
Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 197, 22–35 (2007)
Li, J., Wang, J., Ye, X.: Superconvergence by L 2 −projections for stabilized finite element methods for the Stokes equations. Int. J. Numer. Anal. Model. 6, 711–723 (2009)
Oldroyd, J.: On the formulation of the rheological equations of state. Proc. R. Soc. Lond. A Math. Phys. Sci. 200, 523–541 (1950)
Pani, A., Yuan, J.: Semidiscrete finite element Galerkin approximation to the equations of motion arising in the Oldroyd model. IMA J. Numer. Anal. 25, 750–782 (2005)
Pani, A., Yuan, J., Damazio, P.: On a linearized backward Euler method for the equations of motion of Oldroyd fluids of order one. SIAM J. Numer. Anal. 44, 804–825 (2006)
Silvester, D.: Optimal low-order finite element methods for incompressible flow. Comput. Methods Appl. Mech. Eng. 111, 357–368 (1994)
Silvester, D., Kechkar, N.: Stabilized bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. Comput. Methods Appl. Mech. Eng. 79, 71–87 (1990)
Sobolevskii, P.: Stabilization of viscoelastic fluid motion (Oldroyd’s mathematical model). Differ. Integral Equ. 7, 1597–1612 (1994)
Sun, H., He, Y., Feng, X.: On error estimates of the pressure-correction projection methods for the time-dependent Navier-Stokes equations. Int. J. Numer. Anal. Model. 8, 70–85 (2011)
Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1984)
Wang, K., Feng, M., He, Y.: Two-level stabilized finite element method for the transient Navier-Stokes equations. Int. J. Comput. Math. 87, 2341–2360 (2010)
Wang, K., He, Y., Feng, X.: On error estimates of the penalty method for the viscoelastic flow problem I: time discretization. Appl. Math. Model. 34, 4089–4105 (2010)
Wang, K., He, Y., Feng, X.: On error estimates of the fully discrete penalty method for the viscoelastic flow problem. Int. J. Comput. Math. 88, 2199–2220 (2011)
Wang, K., He, Y., Shang, Y.: Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete Continuous Dyn. Syst. Ser. B 13, 665–684 (2010)
Wang, K., Lin, Y., He, Y.: Asymptotic analysis of the equations of motion for viscoelastic Oldroyd fluid. Discrete Continuous Dyn. Syst. 32, 657–677 (2012)
Wang, K., Shang, Y., Wei, H.: A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations. Comput. Math. Appl. 62, 1814–1827 (2011)
Wang, K., Shang, Y., Zhao, R.: Optimal error estimates of the penalty method for the linearized viscoelastic flows. Int. J. Comput. Math. 87, 3236–3253 (2010)
Zheng, H., Shan, L., Hou, Y.: A quadratic equal-order stabilized method for Stokes problem based on two local Gauss integrations. Numer. Methods Partial Differ. Equ. 26, 1180–1190 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, K., Si, Z. & Yang, Y. Stabilized finite element method for the viscoelastic Oldroyd fluid flows. Numer Algor 60, 75–100 (2012). https://doi.org/10.1007/s11075-011-9512-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9512-3