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Does anyone know where to find an analytic form for the singular values of the finite-difference approximation to the 2D Laplacian, expressed in matrix form for a square grid? This would be for the standard 5-point FD approximation.

I've found the eigenvalues (they are $\sin^2$ functions), but have not seen expressions for the singular values.

Thanks, Tom

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The Laplacian is SPD so the singular values are the same as the eigenvalues (and the eigenvectors are orthogonal). This article explains the eigenvalues and eigenvectors of the discrete Laplacian obtained using centered differences.

http://en.wikipedia.org/wiki/Kronecker_sum_of_discrete_Laplacians

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