Strictly Convergent Algorithm for an Elliptic Equation With Nonlocal and Nonlinear Boundary Conditions

Abstract

Abstract. The paper describes a formally strictly convergent algorithm for solving a class of elliptic problems with nonlinear and nonlocal boundary conditions, which arise in modeling of the steady-state conductive-radiative heat transfer processes. The proposed algorithm has two levels of iterations, where inner iterations by means of the damped Newton method solve an appropriate elliptic problem with nonlinear, but local boundary conditions, and outer iterations deal with nonlocal terms in boundary conditions.

Convergence of the Finite Volume Method for a Conductive-Radiative Heat Transfer Problem

“…Therefore, since ψ(a + ) = 4d 3 1 a + − 3d 4 1 for a + > d 1 , then the last inequality will yield a + ≤ d 1 , what contradicts with the starting assumption.…”

“…is uniformly Lipschitz continuous and one can apply various damped Newton methods (see, for instance, [3] for the continuous case) to obtain unique solvability of the equation F (w) = f .…”

“…The main interest of the paper is the applicability of the finite volume method to problems with nonlocal and nonlinear boundary conditions for a linear elliptic equation of the divergence type. As a rule, given class of boundary value problems arise in mathematical modeling of conductive-radiative heat transfer, see, for instance, [1,2,3,4,8,9].…”

“…For continuous case these difficulties are tackled via embedding theorems in Sobolev spaces, see, for instance [9], or specific trial functions, see, for instance [1,2,3], which unlikely have discrete analogues.…”