I am trying to solve following 2D Poisson's equation numerically: $$ \Delta \Phi = \rho $$ When $\rho$ and $\Phi$ are periodic in both directions, this can be solved straightforward thanks to Fourier Transforms. Indeed, one only needs to perform the inverse Fourier transform of next equation: $$ \hat{\Phi}_k = - \frac{\hat{\rho}_k}{k^2} $$
However in my current setup, both functions are in reality "shear periodic". In other words, both functions satisfy: $$ f(x,y) = f(x+L_x, y-\alpha) $$ Naively, I would say that the Fourier transform of $\Phi$ and $\rho$ is: $$ \hat{f}_k=\exp(-2i\pi k_x \alpha) \hat{f}_k^P $$ where $f^P$ is the fully periodic function (in other words without the shear). However, at the end of the day the exponential term cancels out on both sides of the equation and the equation in Fourier space remains unchanged. For sure, something is wrong because the shift breaks the periodicity in x direction and therefore I cannot apply Fourier transforms. I suspect that somehow the wave vectors should be affected but I do not know how ...
Therefore, I was wondering if you can help me to find out what is the correct way for solving Poisson's equation with this shift ?
Thank you for your time