In this paper we present the numerical analysis of a three-field stabilized finite element formulation recently proposed to approximate viscoelastic flows. The three-field viscoelastic fluid flow problem may suffer from two types of numerical instabilities: on the one hand we have the two inf-sup conditions related to the mixed nature problem and, on the other, the convective nature of the momentum and constitutive equations may produce global and local oscillations in the numerical approximation. Both can be overcome by resorting from the standard Galerkin method to a stabilized formulation. The one presented here is based on the subgrid scale concept, in which unresolvable scales of the continuous solution are approximately accounted for. In particular, the approach developed herein is based on the decomposition into their finite element component and a subscale, which is approximated properly to yield a stable formulation. The analyzed problem corresponds to a linearized version of the Navier–Stokes/Oldroyd-B case where the advection velocity of the momentum equation and the non-linear terms in the constitutive equation are treated using a fixed point strategy for the velocity and the velocity gradient. The proposed method permits the resolution of the problem using arbitrary interpolations for all the unknowns. We describe some important ingredients related to the design of the formulation and present the results of its numerical analysis. It is shown that the formulation is stable and optimally convergent for small Weissenberg numbers, independently of the interpolation used.
Accepté le :
DOI : 10.1051/m2an/2016068
Mots clés : Stabilized finite element methods, viscoelastic fluids, Oldroyd-B, inf-sup conditions
@article{M2AN_2017__51_4_1407_0,
author = {Castillo, Ernesto and Codina, Ramon},
title = {Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1407--1427},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016068},
zbl = {1393.76008},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016068/}
}
TY - JOUR AU - Castillo, Ernesto AU - Codina, Ramon TI - Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1407 EP - 1427 VL - 51 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016068/ DO - 10.1051/m2an/2016068 LA - en ID - M2AN_2017__51_4_1407_0 ER -
%0 Journal Article %A Castillo, Ernesto %A Codina, Ramon %T Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1407-1427 %V 51 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016068/ %R 10.1051/m2an/2016068 %G en %F M2AN_2017__51_4_1407_0
Castillo, Ernesto; Codina, Ramon. Numerical analysis of a stabilized finite element approximation for the three-field linearized viscoelastic fluid problem using arbitrary interpolations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1407-1427. doi : 10.1051/m2an/2016068. http://www.numdam.org/articles/10.1051/m2an/2016068/
F.P.T. Baaijens, M.A. Hulsen and P.D. Anderson, The Use of Mixed Finite Element Methods for Viscoelastic Fluid Flow Analysis. John Wiley & Sons, Ltd (2004).
and , Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds. Numer. Math. 63 (1992) 13–27. | DOI | Zbl
, and , Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 104 (1993) 31–48. | DOI | Zbl
R.B. Bird, R.C. Amstrong and O. Hassager. Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics. Wiley, New York, 2nd edition (1987).
R.B. Bird, R.C. Amstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol. 2: Kinetic Theory. Wiley, New York, 2nd edition (1987).
, , and , 3d viscoelastic analysis of a polymer solution in a complex flow. Comput. Methods Appl. Mech. Eng. 180 (1999) 413–430. | DOI | Zbl
and , A continuous interior penalty method for viscoelastic flows. SIAM J. Sci. Comput. 30 (2007) 1156–1177. | DOI | Zbl
, and , Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math. 107 (2007) 213–255. | DOI | MR | Zbl
, and , GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Methods Appl. Mech. Eng. 190 (2001) 3893–3914. | DOI | Zbl
F. Brezzi and M. Fortin, Variational formulations and finite element methods. In Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Series Comput. Math. edited by F. Brezzi and M. Fortin. Springer New York (1991) 1–35. | Zbl
, and , Approximation of the two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched finite element shape functions. J. Non-Newtonian Fluid Mech. 225 (2015) 37–53. | DOI
and , Stabilized stress-velocity-pressure finite element formulations of the Navier-Stokes problem for fluids with non-linear viscosity. Comput. Methods Appl. Mech. Eng. 279 (2014) 554–578. | DOI | Zbl
and , Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem. Comput. Methods Appl. Mech. Eng. 279 (2014) 579–605. | DOI | Zbl
and , First, second and third order fractional step methods for the three-field viscoelastic flow problem. J. Comput. Phys. 296 (2015) 113–137. | DOI | Zbl
R.P. Chhabra and J.F. Richardson, In Non-Newtonian Flow and Applied Rheology. Elsevier Butterworth-Heinemann, 2nd edition (2008).
. Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput. Methods Appl. Mech. Eng. 190 (2000) 1579–1599. | DOI | Zbl
, Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Eng. 191 (2002) 4295–4321. | DOI | Zbl
, Finite element approximation of the three-field formulation of the Stokes problem using arbitrary interpolations. SIAM J. Numer. Anal. 47 (2009) 699–718. | DOI | Zbl
and , A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput. Methods Appl. Mech. Eng. 143 (1997) 373–391. | DOI | Zbl
, and , Subscales on the element boundaries in the variational two-scale finite element method. Comput. Methods Appl. Mech. Eng. 198 (2009) 838–852. | DOI | Zbl
, and , Analysis of the Oseen-viscoelastic fluid flow problem. J. Non-Newtonian Fluid Mechanics 127 (2005) 157–168. | DOI | Zbl
and , Approximation of time-dependent viscoelastic fluid flow: SUPG approximation. SIAM J. Numer. Anal. 41 (2003) 457–486. | DOI | Zbl
, and , Galerkin/least-square finite-element methods for steady viscoelastic flows. J. Non-Newtonian Fluid Mechanics 84 (1999) 233–256. | DOI | Zbl
and , Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mechanics 123 (2004) 281–285. | DOI | Zbl
E. Fernández−Cara, F. Guillén and R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind. in Handbook of Numerical Analysis, VIII. North-Holland (2002). | Zbl
and , Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal.: Theory, Methods Appl. 15 (1990) 849–869. | DOI | Zbl
, Computation of viscoelastic fluid flows using continuation methods. J. Comput. Appl. Math. 225 (2009) 187–201. | DOI | Zbl
, , and , The variational multiscale method-a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166 (1998) 3–24. | DOI | Zbl
, and , Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newtonian Fluid Mech. 127 (2005) 27–39. | DOI | Zbl
and , On Korn’s first inequality for quadrilateral nonconforming finite elements of first order approximation properties. Int. J. Numer. Anal. Model. 2 (2005) 439–458. | Zbl
and , A three-field formulation for incompressible viscoelastic fluids. Special Issue in Honor of K.R. Rajagopal. Inter. J. Eng. Sci. 48 (2010) 1413–1432. | DOI | Zbl
, A multigrid method for viscoelastic fluid flow. SIAM J. Numer. Anal. 42 (2004) 109–129. | DOI | Zbl
and , Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. 21 (2000) 131–146. | DOI | Zbl
and , A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian Fluid Mech. 26 (1987) 77–114. | DOI | Zbl
and , Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows. ESAIM: M2AN 35 (2001) 879–897. | DOI | Numdam | MR | Zbl
, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations. ZAMM J. Appl. Math. Mech. 65 (1985) 449–451. | DOI | Zbl
M. Renardy, Mathematical Analysis of Viscoelastic Flows. CBMS-NSF Regional Conference Series in Applied Mathematics (1989). | Zbl
, Asymptotic structure of the stress field in flow past a cylinder at high Weissenberg number. J. Non-Newtonian Fluid Mech. 90 (2000) 13–23. | DOI | Zbl
M. Renardy, W. Hrusa and J. Nohel, In Mathematical Problems in Viscoelasticity. John Wiley & Sons, Inc., New York (1987). | Zbl
, 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) 362–377. | DOI | Zbl
Cité par Sources :
