Multigrid analysis for the time dependent Stokes problem

Certain implicit time stepping procedures for the incompressible Stokes or Navier-Stokes equations lead to a singular-perturbed Stokes type problem at each type step. The paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method. Several smoothing iterative methods are considered: a smoother based on distributive iterations, the Braess-Sarazin and inexact Uzawa smoother. Convergence analysis is based on smoothing and approximation properties in special norms. A robust (independent of time step and mesh parameter) estimate is proved for the two-grid and multigrid W-cycle convergence factors.