5.3.3 Fourier Transform Method (FT)

Let $u_{0,l}=u_{M,l}$ and $u_{k,0}=u_{k,N}$ (periodic boundary conditions.) Then we may write

$$u_{k,l} = \frac{1}{MN} \sum_{m=0}^{M-1}\sum_{n=0}^{N-1} U_{m,n} e^{-2\pi i \,km/M} e^{-2\pi i \,nl/N}$$

with

$$U_{m,n} = \sum_{k=0}^{M-1}\sum_{l=0}^{N-1} u_{k,l} e^{2\pi i \,km/M} e^{2\pi i \,nl/N}$$

A similar expansion is used for the charge density $\rho_{k,l}$:

$$R_{m,n} = \sum_{k=0}^{M-1}\sum_{l=0}^{N-1} \rho_{k,l} e^{2\pi i \,km/M} e^{2\pi i \,nl/N}$$

Inserting these expressions in

$$u_{k+1,l} -2u_{k,l}+ u_{k-1,l}+u_{k,l+1}-2u_{k,l}+u_{k,l-1} = - (\Delta l)^{2} \rho_{k,l}$$

we find

$$U_{m,n} = \frac{-R_{m,n} (\Delta l)^{2}} {2 [\cos \, 2\pi m/M + \cos \, 2\pi n/N -2]}$$

Use Fast Fourier Transform! ($N \, \ln \, N$ operations instead of $N^{2}$.)

Variants of the method cover other than periodic boundary conditions.