FFT Poisson Solver for non-uniform grid

Question

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.

Answer 1

First, there are non-uniform mesh FFT variations that you could use without having to do the coordinate transformation.

Second, the FFT is not easily applicable to the transformed problem. The reason why the FFT is easily applicable to the Laplace equation without transformation is that (showing this for the 1d case here) $$ {\cal F}[p_{xx}](k) = -k^2 {\cal F}[p](k), $$ i.e., the derivative simply yields a factor under transformation. Consequently, the equation $-p_{xx}=f$ gives you $$ k^2 {\cal F}[p](k) = {\cal F}[f](k), $$ and thus $$ {\cal F}[p](k) = \frac{1}{k^2}{\cal F}[f](k), $$ which is easily invertible to $$ p(x) = {\cal F}^{-1}\left[\frac{1}{k^2}{\cal F}[f](k)\right](x). $$ But if, on the left hand side, you start with $g(\xi)p_{\xi\xi}(\xi)$, then things are not that simple because $$ {\cal F}[g(\xi)p_{\xi\xi}(\xi)] \neq k^2 {\cal F}[g](k){\cal F}[p](k) $$ or anything similarly simple that would allow you to isolate ${\cal F}[p](k)$ to one side.