Existence and approximation of a (regularized) Oldroyd-B model

Two finite element approximations of the Oldroyd-B model for dilute polymeric fluids are considered, in bounded 2- and 3-dimensional domains, under no flow boundary conditions. The pressure and the symmetric conformation tensor are aproximated by either (a) piecewise constants or (b) continuous piecewise linears, the velocity by (a) continuous piecewise quadratics or a reduced version with linear tangential component on each edge, and (b) by continuous piecewise quadratics or the mini-element. Both schemes (a) and (b) 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], where 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 (b) in the presence of an additional dissipative term in the stress equation and a cut-off on the conformation tensor on certain terms like in [Barrett and S\"uli, M3AS 18 (2008) 935--971] for the FENE dumbbell model, we show (subsequence) convergence towards global-in-time weak solutions (when d=2, cut-offs can be replaced with a time step restriction dependent on the spatial discretization parameter). Hence, we prove existence of global-in-time weak solutions to these regularized models.