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We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is c (x) = 1 + Kδ (x − ξ). K = const > 0 represents the magnitude of the heat capacity concentrated at the point x = ξ. An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Subject Classification (1991): 65M06, 65M12, 65M15 Mathematics

-In this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes on nonuniform meshes. As a model problem we consider the first boundary value problem for the Poisson equation. We assume that the solution of the problem belongs to the Sobolev space W s 2 , 2 s 3. Using the interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of data.

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Abstract. First boundary value problem for elliptic equation with youngest coefficient containing Dirac distribution concentrated on a smooth curve is considered. For this problem a finite difference scheme on a special quasiregular grid is constructed. The finite difference scheme converges in discrete W 1 2 norm with the rate O(h 3/2 ). Convergence rate is compatible with the smoothness of input data.