The rates of convergence of the finite difference schemes on nonuniform meshes for parabolic equation with variable coefficients and weak solutions

The convergence of the difference schemes of the second order of local approximation on space for a onedimensional heat conduction equation with variable factors on an arbitrary nonuniform grid is investigated. For the schemes with averaged coefficient of a thermal conduction and averaged right part the evaluations of a rate of convergence in a grid norm $L_2$, agreed with a smoothness of a solution of a boundary value problem are obtained.