\subsection{Coupling the FMM to a mesh for periodic long-range forces}
\label{ssec:mesh_summary}

\begin{equation}
  S(x) = \frac{e^x}{1 + e^x}
\end{equation}

\begin{align}
  \varphi_s(r) &= \frac{1}{r}\left[2 - 2S\left(\frac{2r}{r_s}\right)\right] \nonumber\\
  &= \frac{1}{r}\left[2 - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}}\right] 
\end{align}
\begin{align}
  |\mathbf{f}_s(r)| &= \frac{1}{r^2}\left[\frac{4r}{r_s}S'\left(\frac{2r}{r_s}\right) - 2S\left(\frac{2r}{r_s}\right) + 2\right] \nonumber \\
  &= \frac{1}{r^2}\left[\frac{4r}{r_s}\frac{e^{\frac{2r}{r_s}}}{(1+e^{\frac{2r}{r_s}})^2} - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}} + 2\right]
\end{align}

\begin{equation}
  \tilde\varphi_l(k) = \frac{1}{k^2}\left[\frac{\upi}{2}kr_s\textrm{csch}\left(\frac{\upi}{2}kr_s\right) \right]
\end{equation}

\begin{figure}
\includegraphics[width=\columnwidth]{potential_short.pdf}
\caption{aa}
\label{fig:fmm:potential_short}
\end{figure}


\begin{figure}
\includegraphics[width=\columnwidth]{force_short.pdf}
\caption{bb}
\label{fig:fmm:force_short}
\end{figure}


\begin{figure}
\includegraphics[width=\columnwidth]{potential_long.pdf}
\caption{cc}
\label{fig:fmm:potential_long}
\end{figure}
