Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part II: A linear scheme

Lukáčová–Medvid’ová, Mária; Mizerová, Hana; Notsu, Hirofumi; Tabata, Masahisa

doi:10.1051/m2an/2017032

Lukáčová–Medvid’ová, Mária ¹ ; Mizerová, Hana ¹ ; Notsu, Hirofumi ^2,³ ; Tabata, Masahisa ⁴

¹ Institute of Mathematics, University of Mainz, Mainz 55099, Germany
² Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan
³ Japan Science and Technology Agency (JST), PRESTO, Kawaguchi 332-0012, Japan
⁴ Department of Mathematics, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo 169-8555, Japan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1663-1689.

Résumé

This is the second part of our error analysis of the stabilized Lagrange–Galerkin scheme applied to 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 leads to an efficient computation with a small number of degrees of freedom especially in three space dimensions. In this paper, Part II, we apply a semi-implicit time discretization which yields the linear scheme. We concentrate on the diffusive viscoelastic model, i.e. in the constitutive equation for time evolution of the conformation tensor a diffusive effect is included. Under mild stability conditions we obtain error estimates with the optimal convergence order for the velocity, pressure and conformation tensor in two and three space dimensions. The theoretical convergence order is confirmed by numerical experiments.

Reçu le : 2016-03-22
Accepté le : 2017-06-19

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/m2an/2017032

Classification : 65M12, 65M25, 65M60, 76A10
Mots-clés : Error estimates, Peterlin viscoelastic model, Lagrange–Galerkin method, Pressure-stabilization

Affiliations des auteurs :

Lukáčová–Medvid’ová, Mária ¹ ; Mizerová, Hana ¹ ; Notsu, Hirofumi ^{2, 3} ; Tabata, Masahisa ⁴

@article{M2AN_2017__51_5_1663_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} {II:} {A} linear scheme},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1663--1689},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {5},
     year = {2017},
     doi = {10.1051/m2an/2017032},
     mrnumber = {3731545},
     zbl = {1421.76160},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2017032/}
}

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 II: A linear scheme
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 1663
EP  - 1689
VL  - 51
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2017032/
DO  - 10.1051/m2an/2017032
LA  - en
ID  - M2AN_2017__51_5_1663_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 II: A linear scheme
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 1663-1689
%V 51
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2017032/
%R 10.1051/m2an/2017032
%G en
%F M2AN_2017__51_5_1663_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 II: A linear scheme. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1663-1689. doi : 10.1051/m2an/2017032. http://www.numdam.org/articles/10.1051/m2an/2017032/

Bibliographie
Cité par

M. Aboubacar, H. Matallah and M.F. Webster, 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

M.A. Alves, P.J. Oliveira and F.T. Pinho, Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newtonian Fluid Mech. 110 (2003) 45–75. | DOI | Zbl

J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow. Numer. Math. 63 (1992) 13–27. | DOI | MR | Zbl

J.W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type models. Math. Models Methods Appl. Sci. 22 (2012) 1150024. | DOI | MR | Zbl

A.V. Bhave, R.C. Armstrong and R.A. Brown, Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions. J. Chem. Phys. 95 (1991) 2988–3000. | DOI

A. Bonito, P. Clément and M. Picasso, Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math. 107 (2007) 213–255. | DOI | MR | Zbl

A. Bonito, M. Picasso and M. Laso, Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys. 215 (2006) 691–716. | DOI | MR | Zbl

J. Bonvin, M. Picasso and R. Stenberg, GLS and EVSS methods for a three-field Stokes porblem arising from viscoelastic flows. Comput. Methods Appl. Mech. Eng. 190 (2001) 3893–3941. | DOI | MR | Zbl

K. Boukir, Y. Maday, B. Métivet and E. Razafindrakoto, A high-order characteristics/finite element method for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 25 (1997) 1421–1454. | DOI | MR | Zbl

S. Boyaval, T. Lelièvre and C. Mangoubi, 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

F. Brezzi and J. Douglas Jr. 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. Vieweg, Wiesbaden (1984) 11–19. | MR | Zbl

P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

M.J. Crochet and R. Keunings, Finite element analysis of die swell of a highly elastic fluid. J. Non-Newtonian Fluid Mech. 10 (1982) 339–356. | DOI | Zbl

J. De Frutos and B. García–Archilla, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements. J. Scientific Comput. 66 (2016) 991–1024. | DOI | MR | Zbl

P. Degond and H. Liu, Kinetic models for polymers with inertial effects. Netw. Heterogen. Media 4 (2009) 625–647. | DOI | MR | Zbl

Y. Fan, A comparative study of the discontinuous Galerkin and continuous SUPG finite element methods for computation of viscoelastic flows. Comput. Methods Appl. Mech. Eng. 141 (1997) 47–65. | DOI | Zbl

Y. Fan, R.I. Tanner and N. Phan–Thien, Galerkin/least-square finite-element methods for steady viscoelastic flows. J. Non-Newtonian Fluid Mech. 84 (1999) 233–256. | DOI | Zbl

R. Fattal and R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123 (2004) 281–285. | DOI | Zbl

R. Fattal and R. Kupferman, 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

M. Fortin and A. Fortin, A new approach for the FEM simulations of viscoelastic flows. J. Non-Newtonian Fluid Mech. 32 (1989) 295–310. | DOI | Zbl

L.P. Franca and R. Stenberg, Error analysis of some Galerkin least squares methods for the elasticity equations. SIAM J. Nume. Anal. 28 (1991) 1680–1697. | DOI | MR | Zbl

R. Guénette and M. Fortin, A new mixed finite element method for computing viscoelastic flows. J. Non-Newtonian Fluid Mech. 60 (1995) 27–52. | DOI

J.G. Heywood and R. Rannacher. Finite-element approximation of the nonstationary Navier–Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. | DOI | MR | Zbl

R. Keunings, On the high Weissenberg number problem. J. Non-Newtonian Fluid Mech. 20 (1986) 209–226. | DOI | Zbl

Y.-J. Lee and J. Xu, New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput. Methods Appl. Mech. Eng. 195 (2006) 1180–1206. | DOI | MR | Zbl

Y.-J. Lee, J. Xu and C.-S. Zhang, Global existence, uniqueness and optimal solvers of discretized viscoelastic flow models. Math. Models Methods Appl. Sci. 21 (2011) 1713–1732. | DOI | MR | Zbl

M. Lukáčová–Medvid’Ová, H. Mizerová, H. Notsu and M. Tabata, Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part I: A nonlinear scheme. ESAIM: M2AN 51 (2017) 1637–1661. | DOI | Numdam | MR | Zbl

M. Lukáčová–Medvid’Ová, H. Mizerová and Š. Nečasová, Global existence and uniqueness result for the diffusive Peterlin viscoelastic model. Nonlin. Anal. Theory, Methodes Appl. 120 (2015) 154–170. | DOI | MR | Zbl

M. Lukáčová–Medvid’Ová, H. Mizerová, Š. Nečasová and M. Renardy, Global existence result for the generalized Peterlin viscoelastic model. SIAM J. Math. Anal. 49 (2017) 2950–2964. | DOI | MR | Zbl

M. Lukáčová–Medvid’Ová, H. Notsu and B. She, Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid. Int. J. Numer. Methods Fluids 81 (2016) 523–557. | DOI | MR

J. Nečas, Les Méthodes Directes en Théories des Équations Elliptiques. Masson, Paris (1967). | MR | Zbl

J.M. Marchal and M.J. Crochet, 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).

L. Nadau and A. Sequeira, 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

H. Notsu and M. Tabata, Error estimates of stable and stabilized Lagrange–Galerkin schemes for natural convection problems. Preprint (2015). | arXiv | Numdam | MR

H. Notsu and M. Tabata, Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Scientific Comput. 65 (2015) 940–955. | DOI | MR | Zbl

H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM: M2AN 50 (2016) 361–380. | DOI | Numdam | MR | Zbl

M.A. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations. Math. Comput. 73 (2003) 1699–1718. | DOI | MR | Zbl

R.G. Owens and T.N. Philips, Computational Rheology. Imperial College Press (2002). | MR | Zbl

R.G. Owens, C. Chauvière and T.N. Philips. A locally-upwinded spectral technique (LUST) for viscoelastic flows. J. Non-Newtonian Fluid Mech. 108 (2002) 49–71. | DOI | Zbl

A. Peterlin, Hydrodynamics of macromolecules in a velocity field with longitudinal gradient. J. Polymer Sci. Part B: Polymer Lett. 4 (1966) 287–291. | DOI

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. ESAIM: M2AN 35 (2001) 879–897. | DOI | Numdam | MR | Zbl

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. | DOI | Zbl

M. Renardy, Mathematical Analysis of Viscoelastic Flows. In vol. 73 of CBMS-NSF Conference Series in Applied Mathematics. SIAM, New York (2000). | MR | Zbl

M. Renardy, Mathematical analysis of viscoelastic fluids. In Vol. 4 of Handbook of Differential Equations: Evolutionary. Amsterdam, North-Holland (2008) 229–265. | MR | Zbl

M. Renardy, 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

M. Renardy and T. Wang, 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

H. Rui and M. Tabata, A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92 (2002) 161–177. | DOI | MR | Zbl

E. Süli, Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53 (1988) 459–483. | 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. Comput.(2017). | MR

Wang K, A new discrete EVSS method for the viscoelastic flows. Comput. Math. Appl. 65 (2013) 609–615. | DOI | MR | Zbl

P. Wapperom, R. Keunings and V. Legat, The backward-tracking Lagrangian particle method for transient viscoelastic flows. J. Non-Newtonian Fluid Mech. 91 (2000) 273–295. | DOI | Zbl

Cité par Sources :