Abstract

In this article we discuss the solution of the Navier-Stokes equations modelling unsteady incompressible viscous flow, by numerical methods combining operator splitting for the time discretization and finite elements for the space discretization.

The discussion includes the description of conjugate gradient algorithms which are used to solve the advection-diffusion and Stokes type problems produced at each time step by the operator splitting methods.

Introduction and Synopsis

The main goal of this article is to review several issues associated to the numerical solution of the Navier-Stokes equations modelling incompressible viscous flow. The methodology to be discussed relies systematically on variational priciples and is definitely oriented to Galerkin approximations. Also, we shall take advantage of time discretizations by operator splitting to decouple the two main difficulties occuring in the Navier-Stokes model, namely the incompressibility condition Δ u=0 and the advection term (u.Δ)u, u being here the velocity field. The space approximation will be based on finite element methods and we shall discuss with some details the compatibility conditions existing between the velocity and pressure spaces; the practical implementation of these finite element methods will also be addressed.

This article relies heavily on [1]-[7] and does not have the pretention to cover the full field of finite element methods for the Navier-Stokes equations; concentrating on books only, pertinent references in this direction are [8]-[14] (see also the references therein).

This article is organized in sections whose list is given just below.

1. The Navier-Stokes equations for incompressible viscous flow

2. Operator splitting methods for initial value problems. Application to the Navier- Stokes equations

3. Iterative solution of the advection-diffusion sub-problems

4. […]