Eigenvalues and Eigenvectors of the Discrete Laplacian
نویسنده
چکیده
We derive explicit formulas for the eigenvalues and eigenvectors of the Discrete Laplacian on a rectangular grid for the standard finite difference and finite element methods in 1D, 2D, and 3D. Periodic, Dirichlet, Neumann, and mixed boundary conditions are all considered. We show how the higher dimensional operators can be written as sums of tensor products of one dimensional operators, and their eigenvectors can be written as tensor products of one dimensional vectors. We also refer the reader to laplacian.m, open source Matlab code for constructing discrete laplacian matrices along with their eigenvalues and eigenvectors.
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