##
A Finite Volume Scheme for Transient Nonlocal Conductive-Radiative
Heat Transfer, Part 2: Convergence to the Weak Solution

#### Peter Philip, *
IMA*

Convergence is proved for a finite volume scheme for transient nonlinear
heat transport equations coupled by nonlocal interface conditions.
The interface conditions model diffuse-gray radiation
between the surfaces of (both open and closed) cavities. The model is
considered in three space dimensions. The special difficulties of the
problem lie in the radiative nonlocal coupling between surfaces and in
the allowed nonlinear dependence of internal energy and emissivities on
the solution (i.e. temperature). Moreover, at material interfaces, the
internal energy and the (otherwise constant) diffusion coefficient can be
discontinuous.
For each time and space discretization, the finite volume scheme gives
rise to a piecewise constant interpolation. It is shown that, if
finenesses of the time and space discretization tend to 0, then a
subsequence of the corresponding interpolations converges to a weak
solution of the continuous problem. A discrete maximum pinciple allows to
prove a discrete H1-estimate as well as estimates of time and space
translates. Convergence is then based on the Kolmogorov compactness
theorem.