Optimal relaxation parameter for the Uzawa Method

We consider the Uzawa method to solve the stationary Stokes equations discretized with stable finite elements. An iteration step consists of a velocity update un+1 involving the (augmented Lagrangian) operator −ν − ρ∇div with ρ ≥ 0, followed by the pressure update pn+1 = p − ανdiv un+1, the so-called Richardson update. We prove that the inf-sup constant β satisfies β ≤ 1 and that, if σ = 1 + ρν−1, the iteration converges linearly with a contraction factor β2ασ−1 ( 2σ − α provided 0 < α < 2σ . This yields the optimal value α = σ regardless of β.