#
Optimal Relaxation Parameter for the Uzawa Method

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Numer. Math. (to appear).
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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
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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 $\beta$.