Discrete Harmonic Functions and Electric Networks
Discrete harmonic functions have a natural interpretation in the context of electric networks. Interpret the lattice points in a 2D domain as nodes of an electric network connected with resistors, and let
denote the voltage at the point
. Then the function
is a discrete harmonic function; see  for details.
Dirichlet Method of Relaxations
This is an iterative method for computing discrete harmonic functions with given boundary values. Starting with an arbitrary initial function
satisfying the boundary conditions, the method constructs a sequence
of functions defined by
approaches a harmonic function; see [1, p. 16].
Three families of boundary functions are provided for the outer and inner (if selected) boundaries of the domain:
are user-defined parameters.
The values of
at the boundary are random real numbers chosen uniformly from the interval
is a user-defined parameter.
For this Demonstration, we chose to initialize the values of
in the interior of the domain
to random real numbers chosen uniformly from the interval
is a user-defined parameter. Choosing
as 0 is equivalent to initializing the values in the interior of
For continuous functions
defined on some domain
, the integral
denotes the gradient, is called the Dirichlet energy integral. This integral can be interpreted as the potential energy of a system or as a measure of the "energy" of the surface
. The smaller this integral, the flatter the surface. A characteristic property of harmonic functions is that they minimize the Dirichlet energy integral. From among all functions having given values on the boundary of
, the harmonic function satisfying the given boundary conditions is the unique function that minimizes the Dirichlet energy integral.
The discrete analog of the Dirichlet energy integral is the sum over the squares of the differences of values of
on all pairs of neighboring points; that is
where the sum runs over all interior lattice points
. In analogy to the continuous case, discrete harmonic functions are the unique functions that, for given boundary conditions, minimize the discrete Dirichlet energy.
 P. G. Doyle and J. L. Snell, Random Walks and Electric Networks
, Washington, DC: Mathematical Association of America, 1984.