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When trying to solve Poisson equation with multigrid considering periodic boundary conditions (PBC) in all directions:

\begin{equation}\nabla^2\phi=f\end{equation}

where\begin{equation}f=-4\pi\rho\end{equation}

I know that the integral over the volume should be equal to zero. That is:

\begin{equation}\int_VfdV=0\end{equation}Wich in the discrete case reads as\begin{equation}\sum f dxdydz=0\end{equation}

As I said before, if we are dealing with periodic bounday conditions, then it imposes also a condition for the potential, as the potential near some of the walls of the simulation box is equal to the potential on the opposite side.

Should it exist another extra condition?

Thanks

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# Multigrid with PBC

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