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\centerline{\bf FINITE DIFFERENCE SCHEMES ON NONUNIFORM}
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\centerline{\bf MESHES FOR PARTIAL DIFFERENTIAL EQUATIONS}
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\centerline{\bf WITH GENERALIZED SOLUTIONS}
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\centerline{\footnotesize BO\v SKO S. JOVANOVI\'C}
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\centerline{\footnotesize\it University of Belgrade, Faculty of Mathematics}
\centerline{\footnotesize\it Studentski trg 16, POB 550, 11000 Belgrade, Yugoslavia}
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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.
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