Existence and Approximation of a (regularized) Oldroyd-b Model

We consider the finite element approximation of the Oldroyd-B system of equations, which models a dilute polymeric fluid, in a bounded domain D ⊂ R, d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the velocity field with continuous piecewise quadratics and either (a) piecewise constants or (b) continuous piecewise linears for the pressure and the symmetric conformation tensor. We show that both of these 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. M2AN 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 et al. M3AS 18 (2008) 935–971] for the microscopicmacroscopic FENE model of a dilute polymeric fluid, we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, towards 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 we 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.