Comparison of preconditioners for collocation Chebyshev approximation of 2D and 3D generalized Stokes problem

The aim of the paper is concerned with the iterative resolution of the generalized Stokes problem in the framework of collocation Chebyshev approximation. More precisely, we analyze the performance of several preconditioners improving the classical Uzawa method. We first recall that the general Stokes problem (GSP) is an elementary substep of general interest for computing not only incompressible flows but also low Mach number flows. We then remark that performances of the classical Uzawa algorithm for solving the GSP are in fact closely related to the Reynolds number. In order to overcome this dependence, preconditioners are needed. The preconditioners we analyze here are recommended by Fourier analysis of the pressure operator. We additionally give interpretation of the preconditioners in terms of influence (or capacitance) techniques. We give a detailed analysis of the conditioning of the discrete collocation Chebyshev versions of the operators. Numerical comparison is conducted in 2D as well as in 3D rectangular geometry. Our comparative study shows that the fictitious wall permeability (FWP) method is the most efficient preconditioner. If complemented with a suitable pressure filtering, its efficiency still increases.