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

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     title = {Numerical analysis of the {Oseen-type} {Peterlin} viscoelastic model by the stabilized {Lagrange{\textendash}Galerkin} method. {Part} {II:} {A} linear scheme},
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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/

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