A combined conjugate gradient - multi-grid algorithm for the numerical solution of the Stokes problem

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Abstract
We consider an algorithm for the solution of a mixed finite-element approximation of the Stokes equations in a bounded, simply connected domain Ω ⊂ R2. The original indefinite problem for the velocity and pressure can be transformed into an equation Lp = g for the pressure involving a symmetric, positive definite, continuous linear operator L: L2(Ω)/R → L2(Ω)/R. We apply a conjugate gradient algorithm to this equation. Each evaluation of Lp requires the solution of two discrete Poisson equations. This is done approximately using a multigrid algorithm. The resulting iterative process has a convergence rate κ bounded away from one independently of the meshsize. Numerical experiments yield values for κ between 0-8 and 0-93. The generalization of the analysis to other mixed problems is obvious.

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