Numerical Solution of the Navier-Stokes Equations by the Finite Element Method

In the article [AF 1 404] we presented the methodological concepts leading to Navier-Stokes equation modeling of incompressible or compressible, laminar or turbulent fluid flow phenomena. To numerically solve these finite-difference equations, we considered, on the one hand, the current-vorticity formulation, which leads to the resolution of Poisson equations coupled to convection-diffusion equations, and, on the other hand, the velocity-pressure formulation. However, the finite-difference approximation of boundary problems is interesting when the domain Ω is rectangular in shape, but difficult to implement when Ω is of any shape. For this reason, the finite element method is preferred in industry, as it can be used to approximate both the partial differential operators and the domain Ω. We also saw in the article [AF 1 406] the solution of the Navier-Stokes equations by the finite volume method.

For reasons of simplicity of exposition, we shall restrict ourselves in this article to the case of stationary two-dimensional problems; usually we discretize the derivatives with respect to time by finite differences to arrive at either explicit or implicit time schemes, then use the finite element method to spatially discretize the semi-discretized time problem. Whether it's one or the other, the crucial point is the spatial approximation of the operators. Finite element time discretization will also be discussed.

Given the complexity of the finite element approximation of the target problem, we will initially restrict ourselves to the finite element approximation of the linear Stokes problem, corresponding to the Navier-Stokes equations without the nonlinear convection term. Furthermore, to simplify the presentation, we will assume that the boundary conditions are of the homogeneous Dirichlet type, i.e. that $\overrightarrow{U}=\left(u,\mathrm{\hspace{0.17em}}v\right)$...