Finite element approximation of the viscoelastic flow problem: A non-residual based stabilized formulation

•A mixed three-field formulation of viscoelastic flows is presented.•Stabilized formulations based on the VMS framework are proposed.•The benefits of the term-by-term stabilization are explained and tested.•The numerical analysis of the stationary and linearized problem is presented.•The formulation is tested numerically.•The robustness of the method is assessed. In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns σ−u−p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.