Splitting Techniques for the Pseudospectral Approximation of the Unsteady Stokes Equations

A pseudospectral (or collocation) approximation of the unsteady Stokes equations is considered. Using the Uzawa algorithm, the spectral system is decoupled into a Helmholtz equation and an equation with a pseudo-Laplace operator. It is proven that the eigenvalues of the pseudo-Laplacian are real and negative (except one zero eigenvalue belonging to the constant mode). The pseudospectral method avoids spurious modes since the pressure is approximated with lower order (two degrees lower) polynomials than the velocity. Only one grid (no staggered grids!) with the standard Chebyshev Gauss–Lobatto nodes is used for discretization. For the iterative solution of the spectral system, an effective preconditioner is necessary. Here effective finite difference preconditioners for the spectral pseudo-Laplacian are presented.