This paper deals with the numerical simulation of circulating steady isothermal flow for liquids with memory. The single-integral constitutive law is applied, assuming the flow kinematics is known. For the calculation of the extra-stress tensor at a particular position in the fluid one has to do following steps. The past positions of the material point have to be tracked, and the relative deformation history has to be calculated and has to be integrated along the path line. This tends to very long calculation times. In recirculating flow regions, for example, the material points have to be tracked over many circulation loops along the path lines, if the memory of the liquid has a very long range compared with the period of a material point for one circulation loop. In this paper, it is shown that the relative deformation history needs only to be determined for a maximum of two loops for the numerical calculation of stresses at a particular position. In certain circumstances this method is able to reduce the calculation time for the simulation of circulating flows of memory fluids drastically. The discretization of the constitutive equations and the conservation equations of momentum and mass occur on the basis of the mixed finite-element method. This leads to a non-linear system of equations, which is solved by a quasi-Newton iterative scheme. The finite element method is applied to the simulation of the flow through a single-screw extruder where the algorithm for the calculation of stress in circulating flows can be applied, because the flow bears a similarity to circulating flow when observed from the rotating screw. The required calculation time would be at least ten time longer if this algorithm were not applied. The numerical results are compared with experimental data.