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Poisson equation fourier transform before discretization

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Old   November 9, 2010, 12:00
Default Poisson equation fourier transform before discretization
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Yohei Kawazura
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I tried to solove Poisson equation by spectral method.
Referring this note(, I pluged FFT for discretized form as,
(U_j+1,k + U_j-1,k + U_j,k+1, Uj, K-1 - 4U_j,k)/h^2 = -f_j,k.

Fourier transform is defined as,
W = exp(2*i*pi/N)

Then I obtain,
U'_m,n = -h^2*f_j,k/(W^m+W^-m+W^n+W^-n-4).

With inverse transfom, seemingly correct result is obtained.

However, I think fourier transform before discretization is also correct.
I mean,
d^2U/dx^2 + d^2U/dy^2 = -f,
is transformed as,
(-m^2-n^2)U'_m,n = -f'_m,n.

Then, inverse transforming for above equation must return correct result.
However, no correct answer is obtained.

What is missing in the second approach?
And how I can solve it?

Thanks in advance.
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