Upwind discretization of a time-dependent two-dimensional grade-two fluid model

In this paper, we propose a finite-element scheme for solving numerically the equations of a transient two-dimensional grade-two non-Newtonian Rivlin–Ericksen fluid model. This system of equations is considered an appropriate model for the motion of a water solution of polymers. By introducing a new variable denoted zz, we split the problem into a coupled one with a transport equation. As one of our aims is to derive unconditional a priori estimates from the discrete analogue of the transport equation, we stabilize our scheme by adding a consistent stabilizing term. We use the P2−P1P2−P1 Taylor–Hood finite-element scheme for the velocity v and the pressure pp, and the discontinuous P1P1 finite element for an auxiliary variable z. The error is of the order of h3/2+kh3/2+k, considering that the discretization of the transport equation loses inevitably a factor h1/2h1/2.