Kronecker sum of discrete Laplacians

Knowing all eigenvalues and eigenvectors of the factors, all eigenvalues and eigenvectors of the Kronecker product can be explicitly calculated. Based on this, eigenvalues and eigenvectors of the Kronecker sum can also be explicitly calculated.

The eigenvalues and eigenvectors of the standard central difference approximation of the second derivative on an interval for traditional combinations of boundary conditions at the interval end points are well known. Combining these expressions with the formulas of eigenvalues and eigenvectors for the Kronecker sum, one can easily obtain the required answer.

${\displaystyle L=\mathbf {D_{xx}} \otimes \mathbf {I} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {D_{yy}} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {I} \otimes \mathbf {D_{zz}} ,\,}$

where ${\displaystyle \mathbf {D_{xx}} ,\,\mathbf {D_{yy}} }$ and ${\displaystyle \mathbf {D_{zz}} }$ are 1D discrete Laplacians in every of the 3 directions, and ${\displaystyle \mathbf {I} }$ are the identities of appropriate sizes. Each 1D discrete Laplacian must correspond to the case of the homogeneous Dirichlet boundary condition, in order to generate the 3D discrete Laplacian L corresponding to the homogeneous Dirichlet boundary condition everywhere on the boundary. The eigenvalues are

${\displaystyle \lambda _{jx,jy,jz}=-{\frac {4}{h_{x}^{2}}}\sin \left({\frac {\pi j_{x}}{2(n_{x}+1)}}\right)^{2}-{\frac {4}{h_{y}^{2}}}\sin \left({\frac {\pi j_{y}}{2(n_{y}+1)}}\right)^{2}-{\frac {4}{h_{z}^{2}}}\sin \left({\frac {\pi j_{z}}{2(n_{z}+1)}}\right)^{2}}$

where ${\displaystyle j_{x}=1,\ldots ,n_{x},\,j_{y}=1,\ldots ,n_{y},\,j_{z}=1,\ldots ,n_{z},\,}$, and the corresponding eigenvectors are

${\displaystyle v_{ix,iy,iz,jx,jy,jz}={\sqrt {\frac {2}{n_{x}+1}}}\sin \left({\frac {i_{x}j_{x}\pi }{n_{x}+1}}\right){\sqrt {\frac {2}{n_{y}+1}}}\sin \left({\frac {i_{y}j_{y}\pi }{n_{y}+1}}\right){\sqrt {\frac {2}{n_{z}+1}}}\sin \left({\frac {i_{z}j_{z}\pi }{n_{z}+1}}\right)}$

where the multi-index ${\displaystyle {jx,jy,jz}}$ pairs the eigenvalues and the eigenvectors, while the multi-index ${\displaystyle {ix,iy,iz}}$ determines the location of the value of every eigenvector at the regular grid. The boundary points, where the homogeneous Dirichlet boundary condition are imposed, are just outside the grid.

An OCTAVE/MATLAB code http://www.mathworks.com/matlabcentral/fileexchange/27279-laplacian-in-1d-2d-or-3d is available under a BSD License, which computes the sparse matrix of the 1, 2D, and 3D negative Laplacians on a rectangular grid for combinations of Dirichlet, Neumann, and Periodic boundary conditions using Kronecker sums of discrete 1D Laplacians. The code also provides the exact eigenvalues and eigenvectors using the explicit formulas given above.