Geodesic
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method. Part I: A nonlinear scheme
Lukáčová–Medvid’ová, Mária1 ; Mizerová, Hana1 ; Notsu, Hirofumi2, 3 ; Tabata, Masahisa4
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, Saitama 332-0012, Japan
4 Department of Mathematics, Waseda University, Tokyo 169-8555, Japan
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1637-1661

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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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016078
Classification : 65M12, 65M25, 65M60, 76A10
Keywords: Error estimates, Peterlin viscoelastic model, Lagrange–Galerkin method, Pressure-stabilization

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, Saitama 332-0012, Japan
4 Department of Mathematics, Waseda University, Tokyo 169-8555, Japan 
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     title = {Numerical analysis of the {Oseen-type} {Peterlin} viscoelastic model by the stabilized {Lagrange{\textendash}Galerkin} method. {Part} {I:} {A} nonlinear 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 I: A nonlinear scheme. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1637-1661. doi: 10.1051/m2an/2016078

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