A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations. I Theory

The present investigation is concerned with a numerical method which has been developed for solving the time-dependent, incompressible Navier-Stokes equations (or variants, such as the Boussinesq equations or the anelastic equations) and the advection-diffusion equation in two and three dimensions. The technique was originally derived via the conventional Galerkin finite element method. In the current investigation, attention is given to two ensuing simplifying approximations which generate a scheme that is probably better described as a blend of finite elements and finite differences, i.e., an 'isoparametric element, finite difference method'. The governing equations and basic spatial discretization are considered along with a one-point quadrature and its effects, and time integration.