Discrete Laplacian

returns
a discrete approximation of Laplace’s differential
operator applied to `L`

= del2(`U`

)`U`

using the default
spacing, `h = 1`

, between all points.

specifies the spacing `L`

= del2(`U`

,`hx,hy,...,hN`

)`hx,hy,...,hN`

between points in each
dimension of `U`

. Specify each spacing input as a scalar or a
vector of coordinates. The number of spacing inputs must equal the number of
dimensions in `U`

.

The first spacing value

`hx`

specifies the*x*-spacing (as a scalar) or*x*-coordinates (as a vector) of the points. If it is a vector, its length must be equal to`size(U,2)`

.The second spacing value

`hy`

specifies the*y*-spacing (as a scalar) or*y*-coordinates (as a vector) of the points. If it is a vector, its length must be equal to`size(U,1)`

.All other spacing values specify the spacing (as scalars) or coordinates (as vectors) of the points in the corresponding dimension in

`U`

. If, for`n > 2`

, the`n`

th spacing input is a vector, then its length must be equal to`size(U,n)`

.

If the input `U`

is a matrix, the interior
points of `L`

are found by taking the difference
between a point in `U`

and the average of its four
neighbors:

$${L}_{ij}=\left[\frac{\left({u}_{i+1,j}+{u}_{i-1,j}+{u}_{i,j+1}+{u}_{i,j-1}\right)}{4}-{u}_{i,j}\right]\text{\hspace{0.17em}}.$$

Then, `del2`

calculates the values on the edges
of `L`

by linearly extrapolating the second differences
from the interior. This formula is extended for multidimensional `U`

.