\documentstyle[twoside]{article} \begin{document} \normalsize \centerline{\bf FINITE DIFFERENCE SCHEMES ON NONUNIFORM} \smallskip \centerline{\bf MESHES FOR PARTIAL DIFFERENTIAL EQUATIONS} \smallskip \centerline{\bf WITH GENERALIZED SOLUTIONS} \bigskip \centerline{\footnotesize BO\v SKO S. JOVANOVI\'C} \medskip \centerline{\footnotesize\it University of Belgrade, Faculty of Mathematics} \centerline{\footnotesize\it Studentski trg 16, POB 550, 11000 Belgrade, Yugoslavia} \bigskip \bigskip Nonuniform meshes are often used for approximation of problems with generalized solutions. In this case, the order of local approximation is usually reduced. In many papers it is shown that the accuracy of the finite difference schemes can be increased using approximation of the considered differential equation in some non--mesh points. In present paper the convergence of such finite difference schemes is proved in discrete $L_2$ and $W_2^1$ norms assuming that the generalized solution of the considered boundary value problem belongs to the corresponding Sobolev space. Obtained convergence rate estimates are consistent with the smoothness of the solution of boundary value problem. The convergence of considered finite difference schemes is proved using integral representations of residuals and interpolation of function spaces instead of the Bramble--Hilbert lemma, which usually involves unnecessary restrictions on the mesh step sizes. Contrary to the finite difference schemes approximating problems with smooth solutions, where the right hand sides of equations are taken in some intermediate non--mesh points, we replace the right hand sides with some averaged values. This is necessary because in the problems with generalized solutions the right hand sides of equations may be discontinuous functions. For this purpose, several averaging operators are defined and applied. Special attention is given to the multidimensional problems and to the problems with variable coefficients. In the last case, averaging operator of exact finite difference scheme is used. Note that in some cases similar results can be obtained using the finite volume method. \end{document} 