Convergence of finite-difference schemes for Poisson's equation with boundary condition of the third kind

Abstract

In this paper we study the convergence of finite-difference schemes to generalized solutions of the third boundary-value problem for Poisson's equation on the unit square. Using the generalized Bramble-Hilbert lemma, we obtain error estimates in discrete $H^1$ Sobolev norm compatible, in some cases, with the smoothness of the data. The outline of the paper is as follows. In section 1 notational conventions are presented. The stability theorem is proved in section~2. In section 3 we prove estimates of the energy of the operator ${\mathrm{\Delta }}_h$. Finally, in section 4, we derive our main results.