A note on the approximation of mildly nonlinear Dirichlet problems by finite differences

Summary In [9], Simpson proved some theorems concerning the approximation of mildly nonlinear Dirichlet problems –u=f (x, u) inD, u=0 on D by finite differences. The assumptionsf(x, 0)0 andf_u(x, u)>0 in [9] have turned out to be unnecessarily restrictive and are eliminated in this paper. On the other hand, we considered it necessary to make the smoothness conditions forD slightly more stringent irrespective of the conditions imposed onf.The results of this paper are already contained in the author's doctoral thesis [6]. Meanwhile, H.B. Keller (Math. Comp. 29, p. 464–476) has published a general theory on approximation methods for nonlinear problems which can be used for obtaining Theorem 1This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and EURATOM