Optimal Relaxation Parameter for the Uzawa Method

Numer. Math. (to appear).

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA


Jae-Hong Pyo
Department of Mathematics
Purdue University
West Lafayette IN 47907, USA.



We consider the Uzawa method to solve the stationary Stokes equations discretized with stable finite elements. An iteration step consists of a velocity update $\u^{n+1}$ involving the (augmented Lagrangian) operator $-\nu\Delta-\rho\na\div$ with $\rho\ge0$, followed by the pressure update $p^{n+1}=p^n-\al\nu \div \u^{n+1}$, the so-called Richardson update. We prove that the inf-sup constant $\beta$ satisfies $\beta\le1$ and that, if $\sigma=1+\rho\nu^{-1}$, the iteration converges linearly with a contraction factor $\be^2\alpha\sigma^{-1}\big(2\sigma-\alpha\big)$ provided $0<\alpha<2\sigma$. This yields the optimal value $\alpha=\sigma$ regardless of $\beta$.