Discrete Laplace operator

Summary

The discrete Laplace operator is the analogue of the Laplacian for functions on a grid or on the vertices of a graph. At each point it adds up the differences between the values at the neighbours and the value at the point. On the grid $h\Z^d$ it gives the standard finite difference approximation of $\sum_i \partial^2 u / \partial x_i^2$. On a finite graph it is, up to sign, the Laplacian matrix, and it links random walks, electrical networks and discrete harmonic functions.