Homogeneous difference schemes for one-dimensional problems with generalized solutions

Exact and truncated homogeneous difference schemes of arbitrary order of accuracy are constructed and investigated for the one-dimensional second-order equation $(k(x)u'(x))'-q(x)u(x)=-f(x)$, $0<x<1$, with generalized solutions in $W_2^1$. Mathematical tools are developed for studying the accuracy of truncated difference schemes. It is assumed that $k(x)$ is a measurable function, while $q(x)$ and $f(x)$ are generalized derivatives of functions in the class $W_p^\lambda$, $0<\lambda\leqslant1$, $2\leqslant p<\infty$; this allows one to include the case in which $q(x)$ and $f(x)$ are $\delta$-functions. It is shown that truncated schemes of $m$th order have accuracy $O(h^{2(m+1)-n})$, where $h$ is the mesh step size and $n$ a number depending on the exponents $\lambda_q$, $\lambda_f$, $p_q$ and $p_f$. In the case of piecewise smooth coefficients $n=0$, and the estimates obtained coincide with results of the theory of homogeneous difference schemes of Tikhonov and Samarskii.
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