+
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
\right.
\label{eq:02}
\end{equation}
-where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as
+where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
\begin{equation}
-\begin{array}{ll}
-u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
- & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
- & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
- & u^k(x,y,z-1) + u^k(x,y,z+1)),
+\begin{array}{l}
+u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
+%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
+ % & u(x,y-1,z) + u(x,y+1,z) + \\
+ % & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
\end{array}
\label{eq:03}
\end{equation}
-where the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite.
+where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite.
The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries.
\item Maximum number of iterations;
\item Precisions on the residual error;
\item Matrix size $N_x$, $N_y$ and $N_z$;
-\item Matrix diagonal value: \np{1.0} (See~(\ref{eq:03}));
-\item Matrix off-diagonal value: \np{-1}/\np{6} (See~(\ref{eq:03}));
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;
\item Communication mode: asynchronous.
\end{itemize}
