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: