A Preconditioned Conjugate Residual Algorithm for the Stokes Problem

Abstract

We present a preconditioned conjugate residual algorithm for a mixed finite element discretization of the Stokes problem

$$ - \Delta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {u} + \nabla \rho = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {f} {\text{ }}in{\text{ }}\Omega {\text{ }},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {u} = 0{\text{ on }}\partial \Omega {\text{ }}div{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {u} = 0{\text{ }}in{\text{ }}\Omega $$

(1.1)

in a plane polygonal domain Ω. The preconditioning relies on the idea of hierarchical basis functions for finite elements (cf.[7, 8]). The algorithm has a quasi optimal convergence rate of 1-0(|logh|).