Finite element approximation of viscoelastic fluid flow using characteristics method Original Research Article

It is known that for the numerical approximation of Oldroydʹs B model for viscoelastic fluid flows some upwinding is needed for the convection of the extra-stress tensor. In this paper we make the numerical analysis of such an approximation with upwinding by the method of characteristics in a finite element context. The approximate stress, velocity, and pressure are, respectively, P1 discontinuous, P2 continuous, and P1 continuous. We suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. We show by a fixed point method that the approximate problem has a solution and an error bound is given.