A Fast Multilevel Algorithm for the Solution of Nonlinear Systems of Conductive-Radiative Heat Transfer Equations in Two Space Dimensions

In this paper we describe a fast multilevel algorithm for the solution of a system of nonlinear integrodifferential equations that model steady-state combined conductive-radiative heat transfer in two space dimensions. This extends our previous work in one space dimension. We formulate the equations as a compact fixed point problem with the temperature as the unknown. The fixed point map requires both a Poisson solve and a transport solve for its evaluation. As a solver for both the transport problem and the full system we apply the Atkinson-Brakhage algorithm, using Newton-GMRES as the solver on the coarse mesh. We compare our solver choices with Newton-GMRES. Under modest stability and convergence assumptions on the transport solver, we prove convergence of the multilevel method for the complete system.