We present a nonlinear stabilized Lagrange–Galerkin scheme for the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi–Pitkäranta’s stabilization method for the conforming linear elements, which yields an efficient computation with a small number of degrees of freedom. We prove error estimates with the optimal convergence order without any relation between the time increment and the mesh size. The result is valid for both the diffusive and non-diffusive models for the conformation tensor in two space dimensions. We introduce an additional term that yields a suitable structural property and allows us to obtain required energy estimate. The theoretical convergence orders are confirmed by numerical experiments. In a forthcoming paper, Part II, a linear scheme is proposed and the corresponding error estimates are proved in two and three space dimensions for the diffusive model.
Accepté le :
DOI : 10.1051/m2an/2016078
Mots-clés : Error estimates, Peterlin viscoelastic model, Lagrange–Galerkin method, Pressure-stabilization
@article{M2AN_2017__51_5_1637_0,
author = {Luk\'a\v{c}ov\'a{\textendash}Medvid{\textquoteright}ov\'a, M\'aria and Mizerov\'a, Hana and Notsu, Hirofumi and Tabata, Masahisa},
title = {Numerical analysis of the {Oseen-type} {Peterlin} viscoelastic model by the stabilized {Lagrange{\textendash}Galerkin} method. {Part} {I:} {A} nonlinear scheme},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1637--1661},
publisher = {EDP-Sciences},
volume = {51},
number = {5},
year = {2017},
doi = {10.1051/m2an/2016078},
mrnumber = {3731544},
zbl = {1421.76159},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016078/}
}
TY - JOUR AU - Lukáčová–Medvid’ová, Mária AU - Mizerová, Hana AU - Notsu, Hirofumi AU - Tabata, Masahisa TI - Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part I: A nonlinear scheme JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1637 EP - 1661 VL - 51 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016078/ DO - 10.1051/m2an/2016078 LA - en ID - M2AN_2017__51_5_1637_0 ER -
%0 Journal Article %A Lukáčová–Medvid’ová, Mária %A Mizerová, Hana %A Notsu, Hirofumi %A Tabata, Masahisa %T Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part I: A nonlinear scheme %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1637-1661 %V 51 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016078/ %R 10.1051/m2an/2016078 %G en %F M2AN_2017__51_5_1637_0
Lukáčová–Medvid’ová, Mária; Mizerová, Hana; Notsu, Hirofumi; Tabata, Masahisa. Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part I: A nonlinear scheme. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1637-1661. doi : 10.1051/m2an/2016078. http://www.numdam.org/articles/10.1051/m2an/2016078/
, and , Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows. J. Non-Newtonian Fluid Mech. 103 (2002) 65–103. | DOI | Zbl
, and , Polymer-solution rheology based on a finitely extensible bead-spring chain model. J. Non-Newtonian Fluid Mech. 7 (1980) 213–235. | DOI | Zbl
, and , Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math. 107 (2007) 213–255. | DOI | MR | Zbl
, and , Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys. 215 (2006) 691–716. | DOI | MR | Zbl
, and , Free-energy-dissipative schemes for the Oldroyd-B model. ESAIM: M2AN 43 (2009) 523–561. | DOI | Numdam | MR | Zbl
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York, 3rd edition (2008). | MR | Zbl
and Stabilized mixed methods for the Stokes problem. Numer. Math. 53 (1988) 225–235. | DOI | MR | Zbl
F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations. In Efficient Solutions of Elliptic Systems, edited by W. Hackbusch. Wiesbaden. Vieweg (1984) 11–19. | MR | Zbl
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. | Numdam | MR | Zbl
and , Finite element analysis of die swell of a highly elastic fluid. J. Non-Newtonian Fluid Mech. 10 (1982) 339–356. | DOI | Zbl
and , Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123 (2004) 281–285. | DOI | Zbl
and , Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newtonian Fluid Mech. 126 (2005) 23–37. | DOI | Zbl
, On the high Weissenberg number problem. J. Non-Newtonian Fluid Mech. 20 (1986) 209–226. | DOI | Zbl
and , New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1180–1206. | DOI | MR | Zbl
, and , Global existence, uniqueness and optimal solvers of discretized viscoelastic flow models. Math. Models Methods Appl. Sci. 21 (2011) 1713–1732. | DOI | MR | Zbl
J.L. Lions, Quelques Méthodes de Résolutiondes Problèmes aux Limites Non Linéaires. Dunod et Gauthier-Villars, Paris (1969). | MR | Zbl
, , and , Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method, Part II: A linear scheme. ESAIM: M2AN 51 (2017) 1663–1689. | DOI | Numdam | MR | Zbl
, and , Global existence and uniqueness result for the diffusive Peterlin viscoelastic model. Nonlin. Anal.: Theory, Methods Appl. 120 (2015) 154–170. | DOI | MR | Zbl
, and , Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid. Int. J. Numer. Meth. Fluids 81 (2016) 523–55. | DOI | MR
, , and , Global existence result for the generalized Peterlin viscoelastic model. SIAM J. Math. Anal. 49 (2017) 2950–2964. | DOI | MR | Zbl
and , A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian Fluid Mech. 26 (1987) 77–114. | DOI | Zbl
H. Mizerová, Analysis and numerical solution of the Peterlin viscoelastic model. Ph.D. thesis, University of Mainz, Germany (2015).
and , Numerical simulations of shear-dependent viscoelastic flows with a combined finite element-finite volume method. Comput. Math. Appl. 53 (2007) 547–568. | DOI | MR | Zbl
J. Nečas, Les Méthodes Directes en Théories des Équations Elliptiques. Masson, Paris (1967). | MR | Zbl
H. Notsu and M. Tabata, Error estimates of stable and stabilized Lagrange–Galerkin schemes for natural convection problems. Preprint (2015). | arXiv | Numdam
and , Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Sci. Comput. 65 (2015) 940–955. | DOI | MR | Zbl
and , Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM: M2AN 50 (2016) 361–380. | DOI | Numdam | MR | Zbl
, Hydrodynamics of macromolecules in a velocity field with longitudinal gradient. J. Polymer Sci. Part B: Polymer Lett. 4 (1966) 287–291. | DOI
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
M. Renardy, Mathematical Analysis of Viscoelastic Flows. CBMS-NSF Conference Series in Applied Mathematics. SIAM, New York 73 (2000). | MR | Zbl
M. Renardy, Mathematical analysis of viscoelastic fluids. In Vol. 4 of Handbook of Differential Equations: Evolutionary Equations, Amsterdam, North-Holland (2008) 229–265. | MR | Zbl
, The mathematics of myth: Yield stress behaviour as a limit of non-monotone constitutive theories. J. Non-Newtonian Fluid Mech. 165 (2010) 519–526. | DOI | Zbl
and , Large amplitude oscillatory shear flows for a model of a thixotropic yield stress fluid. J. Non-Newtonian Fluid Mech. 222 (2015) 1–17. | DOI | MR
and , A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92 (2002) 161–177. | DOI | MR | Zbl
and , Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100 (2005) 351–372. | DOI | MR | Zbl
M. Tabata and S. Uchiumi, An exactly computable Lagrange–Galerkin scheme for the Navier–Stokes equations and its error estimates. To appear in Math. comp. (2017). DOI: 10.1090/mcom/3222. | MR
R. Temam, Navier–Stokes Equations. North-Holland, Amsterdam (1984). | MR | Zbl
, and , The backward-tracking Lagrangian particle method for transient viscoelastic flows. J. Non-Newtonian Fluid Mech. 91 (2000) 273–295. | DOI | Zbl
Cité par Sources :
