In this paper, order one finite elements together with Galerkin least-squares (GLS) methods are used for solving a three-held Stokes problem arising from the numerical study of viscoelastic hows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and stress calculations is proposed. The link with the modified elastic viscous split stress (EVSS) method studied in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95; R. Guenette, M. Fortin, J. Non-Newtonian Fluid Mech. 60 (1995) 27-52) is presented. Numerical results are in agreement with theoretical predictions, and with those presented in (M. Fortin, R. Guenette, R. Pierre, Comput. Methods Appl. Mech. Engrg. 143 (1997) 79-95). (C) 2001 Elsevier Science B.V. All rights reserved.