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:
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]
The Schur complement is:
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:
So the preconditionner suggested is: