First and second order Galerkin-Lagrange methods for transient viscoelastic flows

We consider an error analysis for the first and the second order Galerkin-Lagrange methods applied to the approximation of the time-dependent viscoelasticity equations with an Oldroyd-B constitutive law. We use the method of characteristics and the finite element methods proposed by Fortin-Esselaoui (1987) for the simulation of viscoelastic fluid flow problems. The Lagrangian form of the constitutive law gives an equation of transport type with source term. Firstly we analyze the method of the first order and give numerical results. Next, we present the numerical analysis of the second order method. The time discretization is based on a variant of the method proposed by Pironneau (1989 and 1992) and the extension of the scheme proposed by Rui-Tabata (2001) for a convection-diffusion problem. We show that the scheme obtained for the Oldroyd-B model is stable and its convergence is of second order in time. Through this study and the analysis of the numerical schemes we want to open new perspectives for the numerical simulation of this type of nonlinear problems.