The heat transfer problem (without generation) is modeled with the
Laplace equation 
, where 
is an unknown function 
. Constraints on are given at the boundary (i.e., the boundary
conditions are specified), and the Laplace equation is (approximately)
solved to determine the value of
for points in the interior of the region.
The boundary conditions may be specified as algebraic-topological 1-chains, while the resulting temperature distribution is expressed as a 2-chain. The following images show the solution of the Laplace equation on this domain (the spider) for two different sets of boundary conditions.
In the first, we define boundary conditions that represent a random temperature distribution at the boundary, and compute the steady state temperature distribution within the volume. The resulting distribution is illustrated as follows:
For the second set of boundary conditions, the temperature on the
boundary is proportional to the angle that the surfaces makes with the
vertical, such as would be the case with a very distance radiant heat
source. The resulting distribution within the volume is illustrated
as follows: 