EXISTENCE AND APPROXIMATION OF A (REGULARIZED) OLDROYD-B MODEL

Abstract

We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain [Formula: see text], d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes satisfy a free energy bound, which involves the logarithm of the conformation tensor, without any constraint on the time step for the backward Euler-type time discretization. This extends the results of Boyaval et al. (Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Math. Model. Numer. Anal.43 (2009) 523–561) on this free energy bound. There a piecewise constant approximation of the conformation tensor was necessary to treat the advection term in the stress equation, and a restriction on the time step, based on the initial data, was required to ensure that the approximation to the conformation tensor remained positive definite. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms in the system, similar to those introduced in Barrett and Süli (Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off, Math. Models Methods Appl. Sci.18 (2008) 935–971) for the microscopic–macroscopic finitely extensible nonlinear elastic model of a dilute polymeric fluid, we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, toward global-in-time weak solutions of this regularized Oldroyd-B system. Hence, we prove existence of global-in-time weak solutions to this regularized model. Moreover, in the case d = 2 carry out this convergence in the absence of cut-offs, but with a time step restriction dependent on the spatial discretization parameter, and hence show existence of a global-in-time weak solution to the Oldroyd-B system with an additional dissipative term in the stress equation.