In this paper an optimal control problem for the elliptic boundary value problem with nonlocal boundary conditions is considered. It is shown that the weak solutions of the boundary value problem depend smoothly on the control parameter and that the cost functional of the optimal control problem is Frechet differentiable with respect to the control parameter.

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.

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 L 2type convergence of discrete solutions.

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In this paper we consider a problem about finding of temperature approximation within a thin material sheet involved in conductive‐radiative heat transfer. As result, we found that temperature within the sheet can be approximated in L 2 norm by solution of a simple nonlinear operator equation. Straipsnyje modeliuojamas temperatūros pasiskirstymas tarp plonu medžiagos lakštu atsižvelgiant i radiacijai laidžios šlumos pernešima. Nustatyta, kad temperatūra tarp lakštu gali būti aproksimuojama L 2 normoje paprastos netiesines operatorines lygties sprendiniais.