Uzawa method
We want to solve the same following discrete problem by the Uzawa method, see for example [RT93]:
\[\begin{eqnarray}
\frac{\rho}{\Delta t}\int_{\Omega}{\mathbf{u}_h^{n+1}\cdot\mathbf{v}_h}
+ \mu\int_{\Omega}{\nabla\mathbf{u}_h^{n+1}\cdot\nabla\mathbf{v}_h}
- \int_{\Omega}{p_h^{n+1}\mathrm{div}(\mathbf{v}_h)}
&&\nonumber\\
= \int_{\Omega}{\textbf{f}^{n+1}\cdot\mathbf{v}_h}
&+& \frac{\rho}{\Delta t}\int_{\Omega}{(\mathbf{u}_h^n\circ\textbf{X}_h^n)\cdot\mathbf{v}_h}\\
\int_{\Omega}{\mathrm{div}(\mathbf{u}_h^{n+1})q_h}
&=& 0
\end{eqnarray}\]
Let \(a:V\times V\rightarrow\mathbb{R}\) and \(b:V\times
Q\rightarrow\mathbb{R}\) be bilinear forms defined by :
\[\begin{eqnarray}
a(\mathbf{u}, \mathbf{v})
&=& \frac{\rho}{\Delta t}\int_{\Omega}{\mathbf{u}^{n+1}\mathbf{v}}
+\mu\int_{\Omega}{\nabla\mathbf{u}^{n+1}\cdot\mathbf{v}}\\
b(\mathbf{v}, q) &=& -\int_{\Omega}{p^{n+1}\mathrm{div}(\mathbf{v})}
\end{eqnarray}\]
\[\begin{eqnarray}
a(\mathbf{u}, \mathbf{v})
&=& \frac{\rho}{\Delta t}\int_{\Omega}{\mathbf{u}^{n+1}\mathbf{v}}
+\mu\int_{\Omega}{\nabla\mathbf{u}^{n+1}\cdot\mathbf{v}}\\
b(\mathbf{v}, q) &=& -\int_{\Omega}{p^{n+1}\mathrm{div}(\mathbf{v})}
\end{eqnarray}\]
And let \(f:V\rightarrow\mathbb{R}\) be a linear functionnal defined by:
\[f(\mathbf{u}, \mathbf{v}) = \int_{\Omega}{\textbf{f}^{n+1}\mathbf{v}}
+ \frac{\rho}{\Delta t}\int_{\Omega}{(\mathbf{u}^n\circ\textbf{X}^n)\mathbf{v}}\]
The discrete problem becomes:
\[\begin{eqnarray}
a(\mathbf{u}_h, \mathbf{v}_h) + b(\mathbf{v}_h, p_h) &=& f(\mathbf{v}_h)\\
b(\mathbf{u}_h, q_h) &=& 0
\end{eqnarray}\]
This problem is equivalent to the algrebraic system:
\[\begin{equation}
\left(
\begin{array}{cc}
A & B^T\\
B & 0
\end{array}
\right)
\left(
\begin{array}{c}
\mathbf{u}_h\\
p_h
\end{array}
\right)
=
\left(
\begin{array}{c}
F\\
0
\end{array}
\right)
\label{eqn:algebraic}
\end{equation}\]
Where A, B and F are respectively the representative matrices of a, b and f.
Let introduce the functionnal:
\[J(\mathbf{u}) = \frac{1}{2}(A,\mathbf{u})-(F, \mathbf{u})\]
Then, the problem [eqn:algebraic] is equivalent to the contrained minimization problem:
\[\left\{
\begin{array}{rcl}
J(\mathbf{u}) & = & \inf_{\mathbf{v}\in X}{J(\mathbf{v})}\\
X & = & \{\mathbf{v}\in\mathbb{R}, B\mathbf{u}=0\}
\label{eqn:minimmization}
\end{array}
\right.\]
We associate the Lagrangian function:
\[\mathcal{L}(\mathbf{u},q) = J(\mathbf{u})-(q,B\mathbf{u})\]
A saddle-point \((\mathbf{u},p)\in V\times\mathbb{R}^+\) of L is characterized by the min-max problem:
\[\forall q\in\mathbb{R}^+,
\mathcal{L}(\mathbf{u},q)\leq\mathcal{L}(\mathbf{u},p)\leq\mathcal{L}(\mathbf{v},p), \forall \mathbf{v}\in
V\]
We assume that functions \(J\) and \(b\) are differentiable and
convex. Since Karush-Kuhn-Tucker conditions are verified, a saddle-point
\((\mathbf{u},p)\) of \(\mathcal{L}\) exists and the problem
[eq:const-min] admits a solution.
To find the saddle-point \((\mathbf{u},p)\), we will solve two easiest problems:
\[\mbox{Find }\mathbf{u}_h^n: A\mathbf{u}_h^n + B^Tp_h^n =F\]
\[G(q) = \inf_{v\in V}{\mathcal{L}(\mathbf{v},q)}\]
Then we search \(p\in\mathbb{R}^+\) which maximize \(G\).
Given that \(G(q)=J(\mathbf{u}_q)-(q,B\mathbf{u}_q)\), by derivation we have
\(\nabla G(q) = -B\mathbf{u}_q\). To solve this
constrained maximization problem we can use the projected gradient method:
\[\mbox{Find }p_h^n:p_h^{n+1}=P_{\mathbb{R}^+}(p_h^n-\rho\nabla G(p_h^n))=\max(0,p_h^n+\rho B\mathbf{u}_h^n)\]