Numerical analysis of the modified EVSS method

In this paper, we present a proof of the stability of a new mixed finite element met!lod introduced recently by the authors [9] within the context of viscoelastic fluids. The mixed formulation is related to the EVSS (Elastic-Viscous-Split-Stress) method proposed by Rajagopalan et al. [14] and is based on the introduction of the rate of deformation tensor as an additional unknown. The proof applies to the Stokes flow and some linearized constitutive equations of slow viscoelastic flows. Existence and uniqueness of the continuous and the discrete problems are derived from a generalized Brezzi-Babuska theory. It is shown that no additional compatibility condition is required between the various variables except the usual one for the velocity and the pressure fields. This result allows to choose low order finite element for the stress. Several numerical experiments on the 4:1 contraction Stokes flow will be presented which will confirm the improved stability obtained with this new formulation.