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

rhn@math.umd.edu

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

pjh@math.purdue.edu

Abstract

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 $\beta$.