An ‘assumed deviatoric stress–pressure–velocity’ mixed finite element method for unsteady,convective, incompressible viscous flow: Part I: Theoretical development

A formulation of a mixed finite element method for the analysis of unsteady, convective, incompressible viscous flow is presented in which: (i) the deviatoric-stress, pressure, and velocity are discretized in each element, (ii) the deviatoric stress and pressure are subject to the constraint of the homogeneous momentum balance condition in each element, a priori, (iii) the convective acceleration is treated by the conventional Galerkin approach, (iv) the finite element system of equations involves only the constant term of the pressure field (which can otherwise be an arbitrary polynomial) in each element, in addition to the nodal velocities, and (v) all integrations are performed by the necessary order quadrature rules. A fundamental analysis of the stability of the numerical scheme is presented. The method is easily applicable to 3-dimensional problems. However, solutions to several problems of 2-dimensional Navier-Stokes' flow, and their comparisons with available solutions in terms of accuracy and efficiency, are discussed in detail in Part II of this paper.