X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/0a43df714a5c16cc2f27439bb28f84c2d1f1db16..9f63b1bdd201bf280fa1aee49eeb60425682273c:/paper.tex

diff --git a/paper.tex b/paper.tex
index 93f215d..9d20e32 100644
--- a/paper.tex
+++ b/paper.tex
@@ -367,19 +367,19 @@ It should also be noticed that both solvers have been executed with the Simgrid
 
 In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
 
-\subsection{3D Poisson}
+\subsection{The 3D Poisson problem}
 
 
-We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
+We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
 \begin{equation}
 \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
 \label{eq:07}
 \end{equation}
-such that
+such that:
 \begin{equation*}
 \phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
 \end{equation*}
-where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that:
 \begin{equation}
 \begin{array}{ll}
 \phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
