# Cahouet-Chabard preconditionner

A practical implementation of the Uzawa algorithm requires the choice of a preconditioner which suggest to take $p^{n+1}$ as a suitable approximation. We use the preconditioner given by Cahouet and Chabard [CC98]:

$-\alpha(\Delta)^{-1}+\mu \textbf{I}$ where $\alpha=\frac{\rho}{\Delta t}$ and $\textbf{I}$ is the identity operator.

Indeed; we remain that the Navier-Stokes equations can be rewritten:

$\begin{eqnarray} \alpha\mathbf{u}^{n+1} - \mu\Delta\mathbf{u}^{n+1} &=& \textbf{f}' - \nabla p^{n+1}\label{eqn:uzawaCC}\\ \mathrm{div}(\mathbf{u}^{n+1}) &=& 0 \end{eqnarray}$

Where $\textbf{f}' = \textbf{f}^{n+1}+\alpha\mathbf{u}^n\circ\textbf{X}^n$

Let $H^{-1}$ be a dual space to $H_0^1$. Consider the linear opertor $\textbf{A}^{-1}$, solution of the problem [eq:ns-rewri]

$\begin{array}{rcl} \textbf{A}^{-1}:H^{-1}(\Omega) & \rightarrow & H_0^1(\Omega)\\ \mathbf{v} & \rightarrow & \textbf{A}^{-1}(\mathbf{v})=(\mu\Delta-\alpha\textbf{I})^{-1}\mathbf{v} \end{array}$

The Schur complement is:

$\textbf{A}_0(\alpha) = \mathrm{div}(\textbf{A}^{-1}\nabla)$

The operator $\textbf{A}_0(\alpha)$ is self-adjoint and positive from $L_0^2$ to $L_0^2$. Now the Uzawa algorithm can be considered as a first order Richardson iteration method with a fixed iterative parameter applied to he equation:

$\textbf{A}_0(\alpha)p = \mathrm{div}(\textbf{A}^{-1}:\textbf{f}')$

So the preconditionner suggested is:

$-\alpha(\Delta)^{-1}+\mu\textbf{I}$