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. As a model problem we consider the first initial-boundary value problem for the heat equation with variable coefficients in a domain $(0,1)^2\times (0,T]$. We assume that the solution of the problem and the coefficients of equation belong to corresponding Sobolev spaces. Using interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of the data.