This paper gives very significant and up-to-date analytical and
numerical results to the three-dimensional heat radiation problem
governed by a boundary integral equation. There are two types of
enclosure geometries to be considered: convex and nonconvex
geometries. The properties of the integral operator of the
radiosity equation have been thoroughly investigated and
presented. The application of the Banach fixed point theorem
proves the existence and the uniqueness of the solution of the
radiosity equation. For a nonconvex enclosure geometries, the
visibility function must be taken into account. For the numerical
treatment of the radiosity equation, we use the boundary element
method based on the Galerkin discretization scheme. As a numerical
example, we implement the conjugate gradient algorithm with
preconditioning to compute the outgoing flux for a
three-dimensional nonconvex geometry. This has turned out to be
the most efficient method to solve this type of problems.