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I am using a fast fourier transform in the x direction and a finite difference scheme in the y. This means the poisson equation becomes

-kx^2p_{i,j}+(((p_{i,j-1})-(2*p_{i,j})+(p_{i,j+1}))/(dy^2))=RHS

with dp/dy=0 at the boundaries being enforced using ghost boundary points

this inverts easily for all wavenumbers except when kx=0 when the matrix is singular.

I was wondering if anyone had any experience dealing with this problem and any standard methods of solving such a problem?

Can you simply set p_{hat}=0 at these points and then convert back to real space?

Thanks