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

¹ Institute of Mathematics, University of Mainz, Mainz 55099, Germany
lukacova@uni-mainz.de; mizerova@uni-mainz.de
² Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan
notsu@se.kanazawa-u.ac.jp
³ Japan Science and Technology Agency (JST), PRESTO, Saitama 332-0012, Japan
⁴ Department of Mathematics, Waseda University, Tokyo 169-8555, Japan
tabata@waseda.jp

Received: 22 April 2016
Revised: 23 October 2016
Accepted: 5 December 2016

Abstract

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.