Gave it a read through. Is not good yet but the information should be there.

21_distributed-memory.tex

@@ -397,14 +397,15 @@ subtrees that are deployed to worker processes.
 Only subtrees that overlap with the computational domain are deployed.
 This constraint can be used to steer work distribution.
 
-\exahype can scale the bounding box such that \texttt{outside\_cells\_left} and/or \texttt{outside\_cells\_right} cells are placed outside of the 
+\exahype can scale the bounding box such that \texttt{outside\_cells\_left} and/or \\
+\texttt{outside\_cells\_right} cells are placed outside of the 
 domain while the latter is still resolved as accurately as specified in the spec file.
 
-Furthermore, there is an option to place exactly \texttt{ranks\_per\_dimension" on the coarse grid}. 
-(Note: This overrules \texttt{outside\_cells\_right} but not \texttt{outside\_cells\_left}. 
+Furthermore, there is an option to place exactly \texttt{ranks\_per\_dimension} on the coarse grid. 
+(Note: This overrules \texttt{outside\_cells\_right} but not \texttt{outside\_cells\_left}.) 
 This feature is particular interesting for weak scaling experiments as it can scale the number of cells per dimension
 of a mesh by arbitrary integers.
-An example is given in Fig. \ref{fig:weak-scaling}:
+An example is given in Fig. \ref{fig:weak-scaling}.
 
 \begin{figure}[h]
 \begin{center}
@@ -415,7 +416,8 @@ An example is given in Fig. \ref{fig:weak-scaling}:
   \includegraphics[width=0.4\textwidth]{sketches/weak-scaling-4.pdf}\hspace{0.1cm}
   \includegraphics[width=0.4\textwidth]{sketches/weak-scaling-8.pdf}
 \end{center} 
-\caption{Weak scaling experiment growing the domain by factors $1$, $2^d$, $4^d$, and $8^d$. \texttt{ranks\_per\_dimension} is chosen 1,2,4,8.}
+\caption{Weak scaling experiment growing the domain by factors $1$, $2^d$, $4^d$, and $8^d$. \texttt{ranks\_per\_dimension} is chosen 1,2,4,8.
+The majority of cells lying outside of the domain will be erased.}
 \label{fig:weak-scaling}
 \end{figure}
 
@@ -426,31 +428,18 @@ is interesting since it can be perfectly distributed among
 $2^d$, $4^d$, and $12^d$ processes.
 
 The same mesh refined by a factor $3^i$, $i \leq 0$ can be constructed if $3^i \times 9$ bounding box outside cells
-are placed to the left of the domain.
-The $36^d$ mesh requires a maximum-mesh-size of $\frac{1}{36} \approx 0.03$.
-The uniformly refined variants must scale this mesh size with a factor $\frac{1}{3^i}$.
+are placed to the left of the domain. The $36^d$ mesh requires a maximum-mesh-size of 
+$\frac{1}{36} \approx 0.03$.
+The uniformly refined variants must scale this mesh size with a factor $3^{-i}$.
 
-Another family of interesting uniform meshes are the ones with $3^i*48^d$ cells that are shifted
+Another family of interesting uniform meshes are the ones with $3^i \times 48^d$ cells that are shifted
 by $3^i \times 3$ cells. They can be created by choosing $\texttt{ranks\_per\_dimension}=16$
 and $\texttt{outside\_cells\_left}=3$. The latter must be multiplied by a factor $3^i$ for any
 additional refinement level to preserve the positioning.
 The $48^d$ mesh requires a maximum-mesh-size of $\frac{1}{48} \approx 0.021$.
-For further uniform refinement, the mesh size must be scaled with a factor $\frac{1}{3^i}$.
-
-\begin{figure}[h]
-\begin{center}
-  \includegraphics[width=0.4\textwidth]{sketches/strong-scaling-grid-223.pdf}\hspace{0.1cm}
-  \includegraphics[width=0.4\textwidth]{sketches/strong-scaling-grid-233.pdf}
-\end{center} 
-\caption{Left: A grid with $36^d$ cells on the coarse grid which allows to use $1$, $2^d$, $4^d$, and $12^d$ ranks
-to uniformly distribute the load.
-Right: A grid with $48^d$ cells on the coarse grid which allows to use $1$, $2^d$ and $16^d$ ranks
-to uniformly distribute the load. A quasi-optimal distribution can be established with $4^d$ ranks.}
-\label{fig:strong-scaling}
-\end{figure}
-
-The code to draw the grids is given below:
+For further uniform refinement, the mesh size must be scaled with a factor $3^{-i}$.
 
+The code used to draw all grids is given below:
 \begin{code}
 \documentclass[tikz,preview,crop,11pt]{standalone}
 \begin{document}
@@ -482,6 +471,21 @@ The code to draw the grids is given below:
 \end{document}
 \end{code}
 
+\begin{figure}[h]
+\begin{center}
+  \includegraphics[width=0.4\textwidth]{sketches/strong-scaling-grid-223.pdf}\hspace{0.1cm}
+  \includegraphics[width=0.4\textwidth]{sketches/strong-scaling-grid-233.pdf}
+\end{center} 
+\caption{Left: A grid with $36^d$ cells on the coarse grid which allows to use $1$, $2^d$, $4^d$, and $12^d$ ranks
+to uniformly distribute the load.
+Right: A grid with $48^d$ cells on the coarse grid which allows to use $1$, $2^d$ and $16^d$ ranks
+to uniformly distribute the load. A quasi-optimal distribution can be established with $4^d$ ranks.
+The majority of cells lying outside of the domain will be erased.}
+\label{fig:strong-scaling}
+\end{figure}
+
+
+
 
 \section{MPI troubleshooting and inefficiency patterns}
 \label{section:mpi:inefficiency-patterns}