On the discrete EVSS method

Abstract This paper is a completion of a previous paper written by two of the authors [M. Fortin, R Guenette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] concerning the DEVSS method, which is a discrete variant of the popular elastic-viscous-split-stress (EVSS) method introduced by [Rajagopalan, R.A. Brown, R.C. Armstrong, Finite element methods for calculation of steady viscoelastic flow using constitutive equations with a Newtonian viscosity, J. Non-Newtonian Fluid Mech. 36 (1990) 159–199] within the context of viscoelastic fluids. The method uses the rate of deformation tensor as an additional variable. In this paper, it is shown that a simple modification of the theoretical framework introduced in the first paper [M. Fortin, R Guenette, R. Pierre, Numerical analysis of the modified EVSS method, Comput. Methods. Appl. Mech. Engrg. 143 (1997) 79–95] allows the treatment of multi-mode constitutive equations of viscoelastic flows as well as that of the velocity gradient introduced as the additional variable instead of the rate of deformation tensor. Moreover, this general framework based on a generalized Brezzi–Babuska theory, allows different discretizations for the stress and the additional variable. The paper presents a general criteria insuring the stability of the method. Three particular mixed finite elements satisfying this criteria are studied. An error analysis is performed on Stokes and viscoelastic flows.