We present numerical results for the finite element (FE) method described in [Comput. Meth. Appl. Mech. Eng. 191 (2002) 5045]. This method is dedicated to the approximation of fluid flows obeying to the upper-convected Oldroyd model (so called Oldroyd-B model). One particularity of this method is to take into account the purely viscoelastic case, so called the upper-convected Maxwell model which is important in practice. Numerical results are given for two kinds of problems: a stagnation flow case and also a flow in an abrupt plane 4-1 contraction. The stagnation flow case provides an interesting test because the solution is known and, furthermore it is not integrable when the parameter of elasticity of the fluid is sufficiently high. (C) 2004 Elsevier B.V. All rights reserved.