A Finite Element Preconditioner for Non-Linear Stokes Equations using a Spectral Element Method

The development of efficient tools for the numericalsimulation of three-dimensional fluid-flow problems have received muchattention over the last decade. Numerous theoretical and algorithmicaldevelopments opened the possibilities of spectral methods to complex flowproblems. For these large-scale simulations iterative methods arenecessary and several techniques are proposed ~\cite{Fisher95,Couzy95}.\\In this work a numerical scheme and iterative solver is presented to solvethe unsteady Navier$-$Stokes for a generalized Newtonian fluid in a complexgeometry. The diffusion coefficient becomes non-linear and hence each timestep a new system of equation has to be solved. The equations areintegrated in time using a splitting procedure and then discretizedspatially by means of a high or! der spectral element method. We propose anumerical solution strategy where a static finite element preconditioner isused to limit the number of required iteration steps. Some results aregiven where the clustering of the eigenvalues is analyzed. Finally resultsare presented for the fluid flow in a static mixer, a complex threedimensional geometry.