In electrostatics, the laplacian of the electrostatic potential $\Delta V(\mathbf{r})$ arising from a charge distribution $\rho(\mathbf{r})$ is
$ \Delta V(\mathbf{r})=-\frac{\rho(\mathbf{r})}{\epsilon_0},$
where $\epsilon_0$ is the vacuum permitivity.
I would like to use Fast Fourier Transforms (FFT) to solve this PDE using the FFTW library. FFTW computes $Y$, the Discrete Fourier Transform (DFT) of a 1D complex array $X$ of size $N$ with this definition for the forward transform: \begin{equation} Y_{k}\triangleq\sum_{n=0}^{N-1}X_{n}e^{-2i\pi nk/N}, \end{equation} and the following definition for the backward (or inverse) transform: \begin{equation} Y_{k}=\sum_{n=0}^{N-1}X_{n}e^{2i\pi nk/N}. \end{equation}
As far as I understand, the sampling interval in these definitions is equal to 1. Also, $FFT^{-1}(FFT(X))=N\cdot X$, which is not usual. The manual says this definition is not unitary and that normalization should be taken into consideration by the user.
My application is in 3 dimensions, with regular sampling in each direction, but the sampling frequency is not the same in all directions ($L_1$, $L_2$, $L_3 \neq 1$).
My question : could you help me get an expression of $\Delta V$ in k-space ? With a simplest definition of FFTs, it would be for instance $-\mathbf{k}^2 V(\mathbf{k})$. Here ... I can't get it.
Sorry if the question is a bit messy. I'm not a mathematician and this is quite new for me.