Editing Discrete Laplace operator (section)

=== Finite differences ===

Approximations of the [[Laplacian]], obtained by the [[finite-difference method]] or by the [[finite-element method]], can also be called '''discrete Laplacians'''.  For example, the Laplacian in two dimensions can be approximated using the [[five-point stencil]] finite-difference method, resulting in

:<math> \Delta f(x,y) \approx \frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}, </math>

where the grid size is ''h'' in both dimensions, so that the five-point stencil of a point (''x'',&nbsp;''y'') in the grid is

:<math>\{(x-h, y), (x, y), (x+h, y), (x, y-h), (x, y+h)\}.</math>

If the grid size ''h'' = 1, the result is the '''negative''' discrete Laplacian on the graph, which is the [[square lattice|square lattice grid]]. There are no constraints here on the values of the function ''f''(''x'', ''y'') on the boundary of the lattice grid, thus this is the case of no source at the boundary, that is, a no-flux boundary condition (aka, insulation, or homogeneous [[Neumann boundary condition]]). The control of the state variable at the boundary, as 
''f''(''x'', ''y'') given on the boundary of the grid (aka, [[Dirichlet boundary condition]]),  is rarely used for graph Laplacians, but is common in other applications.

Multidimensional discrete Laplacians on [[Cuboid#Rectangular cuboid|rectangular cuboid]] [[regular grid]]s have very special properties, e.g., they are [[Kronecker product#Kronecker sum and exponentiation|Kronecker sums]] of one-dimensional discrete Laplacians, see [[Kronecker sum of discrete Laplacians]], in which case all its [[eigenvalue]]s and [[eigenvectors]] can be explicitly calculated.