Optimal relaxation parameter for the Uzawa Method

Summary.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 −νΔ−ρ∇÷ with ρ≥0, followed by the pressure update pn+1=pn−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.