An assumed deviatoric stress-velocity-pressure mixed finite element method for unsteady convective, incompressible viscous flow

The formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented, in which: (1) the deviatoric stress, pressure, and velocity are discretized in each element; (2) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori; (3) the convective acceleration is treated by the conventional Galerkin approach; (4) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitarary polynomial) in each element, in addition to the nodal velocities; and (5) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. Methods of solution of the unsymmetric nonlinear equations for steady flow and the first order time dependent differential equations for the unsteady flow and the first order time dependent differential equations for the unsteady flow are discussed at length. This mixed formulation is easily applicable to three dimensional problems. However, solutions to several problems of two dimensional Navier-Stokes flow.