43 
\end{array} \right] 
\end{array} \right] 
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\label{SADDLEPOINT} 
\label{SADDLEPOINT} 
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\end{equation} 
\end{equation} 
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where $A$ is coercive (assuming $A$ is not depending on $v$ or $p$), selfadjoint linear operator in a suitable Hilbert space, $B$ is the $(1) \cdot$ divergence operator and $B^{*}$ is it adjoint operator (=gradient operator). For more details on the mathematics see references \cite{AAMIRBERKYAN2008,MBENZI2005}. 
where $A$ is coercive (assuming $A$ is not depending on $v$ or $p$), selfadjoint linear operator in a suitable Hilbert space, $B$ is the $(1) \cdot$ divergence operator and $B^{*}$ is it adjoint operator (=gradient operator). 
47 

For more details on the mathematics see references \cite{AAMIRBERKYAN2008,MBENZI2005}. 
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We use iterative techniques to solve this problem: Given an approximation $v$ and $p$ for 
We use iterative techniques to solve this problem: Given an approximation $v$ and $p$ for 
49 
velocity and pressure we perform the following steps in the Uzawa scheme \index{Uzawa scheme} style: 
velocity and pressure we perform the following steps in the Uzawa scheme \index{Uzawa scheme} style: 
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\begin{enumerate} 
\begin{enumerate} 
89 
\end{equation} 
\end{equation} 
90 
with sufficient accuracy to return $q=Bw$. Notice that the residual $r$ is given as 
with sufficient accuracy to return $q=Bw$. Notice that the residual $r$ is given as 
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\begin{equation} \label{STOKES RES } 
\begin{equation} \label{STOKES RES } 
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r= B (v\hackscore{1}  B A^{1} B^{*} dp) = B (v\hackscore{1}  A^{1} B^{*} dp) = B (v\hackscore{1}dv\hackscore{2}) = B v\hackscore{2} 
r= B (v\hackscore{1}  A^{1} B^{*} dp) = B (v\hackscore{1}  A^{1} B^{*} dp) = B (v\hackscore{1}dv\hackscore{2}) = B v\hackscore{2} 
93 
\end{equation} 
\end{equation} 
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so in fact the residual $r$ is represented by the updated velocity $v\hackscore{2}$. This saves the recovery of 
so in fact the residual $r$ is represented by the updated velocity $v\hackscore{2}$. This saves the recovery of 
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$dv\hackscore{2}$ in~\ref{SADDLEPOINT ITER STEP 2} after $dp$ has been calculated as iterative method such as PCG calculate the solution approximations along with the their residual. In PCG the iteration is terminated if 
$dv\hackscore{2}$ in~\ref{SADDLEPOINT ITER STEP 2} after $dp$ has been calculated as iterative method such as PCG calculate the solution approximations along with the their residual. In PCG the iteration is terminated if 