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Poisson equation fourier transform before discretizationHi,
I tried to solove Poisson equation by spectral method. Referring this note(http://www.physics.buffalo.edu/phy41...6/ch6-lec2.pdf), 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, U'_m,n=1/N*Sum_{j=0}^{N-1}(W^-{mj+nk}U_j,k), 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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