We show that the finite volume method rigorously converges to the solution of a conductive-radiative heat transfer problem with nonlocal and nonlinear boundary conditions. To get this result, we start by proving existence of solutions for a finite volume discretization of the original problem. Then, by obtaining uniform boundedness of discrete solutions and their discrete gradients with respect to mesh size, we finally get L2type convergence of discrete solutions.
Birgelis, K., & Raitums, U. (2013). Convergence of the finite volume method for a conductive-radiative heat transfer problem. Mathematical Modelling and Analysis, 18(2), 274-288. https://doi.org/10.3846/13926292.2013.783885
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