A fast multilevel algorithm for the solution of nonlinear systems of conductive-radiative heat transfer equations

In this paper we describe and analyze a fast multilevel algorithm for the solution of a system of nonlinear integro-differential equations that model steady-state combined conductive-radiative heat transfer. This system of equations for radiative intensity and temperature can be formulated as a compact fixed point problem in temperature alone with a fixed point map that requires both a solution of the linear transport equation and the linear heat equation for its evaluation. We obtain an efficient evaluation of the fixed point map by coupling a finite element diffusion solver with a fast transport solver developed by the second author. As a solver we apply a modification of the Atkinson-Brakhage method, with Newton-GMRES as the coarse mesh solver, to the full nonlinear system. We compare our discretization/solver pair with Newton-GMRES and the classical Atkinson-Brakhage algorithm.