These last twenty years, the search of robust and efficient strategies for the numerical resolution of the steady or unsteady incompressible Navier-Stokes equations has been a crucial task giving rise to numerous methods. The major obstacle of these equations lies in the absence of pressure terms in the incompressibility constraint. Several ways have been suggested to overcome this difficulty. The first trend is represented by pressure correction methods like SIMPLE or PISO procedures. The main drawback of such techniques lies in the slowing down of convergence when the number of grid points or the clustering ratios over curvilinear grids increase. A possible cure consists in using non-linear multigrid with sequential pressure correction methods as basis solvers (or smoothers). The second trend is to solve the incompressible Navier-Stokes equations in a locally or fully coupled manner, where momentum and continuity equations are solved simultaneously. Coupled strategies for the resolution of the Navier-Stokes equations in primitive variables allow to develop robust solvers and subsequently to understand the limitations of segregated methods.

A fully coupled method for the resolution of the incompressible Navier-Stokes equations is presented. This method previously used by Deng et al. (1991) employs a cell-centered colocated grid, standard linearization of convection terms, central difference discretization for both convective and diffusive terms and a pressure Poisson equation approach, leading to deduce from the incompressibility constraint an equation for the pressure variable. The originality of this present work is to introduce auxiliary variables --the so-called pseudo-velocities-- to make easier the flux reconstruction step. The resulting structure of the nodal unknowns matrix consists in seven or nineteen bands of sparse blocks. Direct solvers have been used to solve coupled systems but their use for three-dimensional applications is penalized by strong storage limitations. In order to improve the global efficiency of the algorithm by seeking grid independent convergence rates, a non-linear multigrid approach is chosen by implementing a FMG-FAS (Full Multigrid-Full Approximation Scheme) procedure for the resolution of the coupled system. Numerical treatments and implementation aspects of this non-linear procedure are detailed.

Steady laminar lid-driven cavity flows calculations have been performed on three-dimensional geometries to discuss the performances of this approach. The retained test-problems were the three-dimensional versions of the ones investigated by Demirdzic et al. (1992) and Oosterlee et al. (1993) : regular, skewed and L-shaped lid-driven cavities. The non-linear multigrid approach (MGC) is compared with the single grid coupled method (SGC) and decoupled methods based on the sequential PISO algorithm (DC, DC-MG) with different pressure linear solvers respectively Krylov subspace solver and a linear multigrid solver. Computations were performed on cartesian, orthogonal and stretched grids for the regular cubic cavity, cartesian and non-orthogonal grids for the skewed lid-driven cavity (skewness angle : 30 degrees) and finally curvilinear grids for the L-shaped cavity. The chosen Reynolds numbers (Re) are equal to 100 or 400. From the whole numerical results, the main conclusion is that the non-linear multigrid approach seems quite powerful, at least more efficient than decoupled or single grid coupled methods.