Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds

We study a finite element approximation of viscoelastic fluid flow obeying an Oldroyd B type constitutive law. The approximate stress, velocity and pressure are respectivelyP 1 discontinuous,P 2 continuous,P 1 continuous. We use the method of Lesaint-Raviart for the convection of the extra stress tensor. 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 we give an error bound.