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I am currently trying to solve the 3D Poisson equation with a Chebyshev discretisation in the $z$ direction (from -1 to 1) and Fourier in the $x$ and $y$ (from $-\pi$ to $\pi$)

I have taken the code into a point where I think it is working but am not totally sure it is outputting the correct result. Does anyone know of any tests that I could use to either confirm it is working or reveal a mistake in it?, e.g., any initial conditions I could insert to make sure the correct result is being obtained.

Thanks for any help!

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It is so trivial to pick a solution on a box domain for the Laplace equation. Just pick a function, say $\bar u(x,y,z)=x^2y^2\sin(z)$ (chosen in a way so that it isn't in your ansatz space), then compute $f=-\Delta \bar u$ and solve the Laplace equation $$ -\Delta u = f. $$ Of course, your numerical solution should converge to $\bar u$. If you can't deal with nonzero boundary conditions, choose some other $\bar u$ that satisfies the boundary conditions you can deal with.

This technique is called the "method of manufactured solutions". Here is another description of it.

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