I have a 3D solver for the incompressible Navier-Stokes equations which uses a FFT library for the Poisson equation with a uniform grid on all directions. In 2D the Poisson equation is given by:
$$ p_{xx} + p_{yy} = f_{rhs} $$
When using a non-uniform grid, we usually map the domain to a computational space where the grid is uniform. Let $x = x(\xi), y = y(\eta)$. The Poisson equation on the computational grid is:
$$ p_{\xi \xi} \, \xi_{x}^2 + p_{\eta \eta} \, \eta_{y}^2 + p_{\xi}\, \xi_{xx} + p_{\eta} \, \eta_{yy} = f_{rhs} $$
Although I do have a background in numerical methods, my knowledge of FFT methods is limited. Is the Poisson equation on a non-uniform grid solvable with FFT methods? If not, what are the alternatives? Note that for reasons that take some time to explain, multigrid and classic iterative methods (e.g. jacobi...) are not an option. Or rather they were shown unstable for certain applications.