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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/m2an/2016078/
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