Abstract.In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. NonNewtonian Fluid Mech. 123 (2004) 281-285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al. J. Non-Newtonian Fluid Mech. 127 (2005) 27-39]). Our analysis gives some tracks to understand these numerical observations. Mathematics Subject Classification. 65M12, 76M10, 35B45, 76A10, 35B35.

We derive a criterion for the breakdown of solutions to the Oldroyd-B model in R 3 in the limit of zero Reynolds number (creeping flow). If the initial stress field is in the Sobolev space H m (R 3 ), m > 5/2, then either a unique solution exists within this space indefinitely, or, at the time where the solution breaks down, the time integral of the L ∞ -norm of the stress tensor must diverge. This result is analogous to the celebrated Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of incompressible fluids. with initial condition ω(·, 0) = ω 0 . Here ω = ∇ × u is the vorticity and u is the velocity field. The BKM theorem states that if ω 0 belongs to the Sobolev space H m (R 3 ), m > 3 2 , then either there exists a solution ω(·, t) ∈ H m (R 3 ) for all times, or, if T * is the maximal time of existence of a solution in H m (R 3 ), BEALE-KATO-MAJDA CRITERION FOR AN OLDROYD-B FLUID 3 then lim tրT * t 0 ω(·, s) L ∞ ds = ∞, and in particular, lim sup tրT * ω(·, t) L ∞ = ∞.That is, the breakup of solutions in any Sobolev norm necessitates the divergence of the L ∞ -norm of the vorticity. The practical implication of this theorem is that breakdown cannot be attributed, say, to the failure of some high derivative. The blowup of the vorticity itself, in the supremum norm, is the signature of any finite-time breakdown. For another criterion of singularity formation see [7]; for up-to-date surveys see [8,9]. The goal of this paper is to establish a similar result for the threedimensional Oldroyd-B model, in the zero-Reynolds number regime. In this regime, a closed equation can be written for the polymeric stress field σ = σ(x, t); this equation is similar to the vorticity equation (1.1). We start by establishing the local-in-time existence of solutions in any Sobolev space H m (R 3 ), m > 5/2. Following then the approach of BKM, we prove that if the initial stress is in H m (R 3 ), then either a solution exists for all time, or, if T * is the maximum existence time, then lim tրT * t 0

a b s t r a c tWe investigate numerical aspects of the Brownian configuration fields method, and in particular its numerical stability as the Weissenberg number increases. Our results show the method to be immune to the type of instability leading to numerical blowup in the simulation of macroscopic models. We discuss this finding in the light of the stability criterion proposed in Fattal et al. [R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg using the log-conformation representation, J. Non Newtonian Fluid Mech. 126 (2005) 23-37].