In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domain.
In a general situation of the separation of variables in the discrete case, the multidimensional discrete Laplacian is a Kronecker sum of 1D discrete Laplacians.
Mathematically, using the Kronecker sum:
L=
Dxx ⊕ Dyy=Dxx ⊗ I+I ⊗ Dyy, |
where
Dxx |
Dyy |
I
Dxx |
Dyy |
Here is a sample OCTAVE/MATLAB code to compute L on the regular 10×15 2D grid:
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 sumcan 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.
L=
Dxx ⊗ I ⊗ I+I ⊗ Dyy ⊗ I+I ⊗ I ⊗ Dzz, |
where
Dxx,Dyy |
Dzz |
I
λjx,jy,jz=-
4 | \sin\left( | |||||
|
\pijx | |
2(nx+1) |
| |||||||||||
\right) | \sin\left( |
\pijy | |
2(ny+1) |
| |||||||||||
\right) | \sin\left( |
\pijz | |
2(nz+1) |
\right)2
where
jx=1,\ldots,nx,jy=1,\ldots,ny,jz=1,\ldots,nz,
vix,iy,iz,jx,jy,jz=\sqrt{
2 | |
nx+1 |
where the multi-index
{jx,jy,jz}
{ix,iy,iz}
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.