Crank–Nicolson's scheme with different time-step in subdomains for the solution of parabolic problems

A method of constructing difference schemes with different time-step in subdomains based on Crank–Nicolson's scheme is suggested. It is associated with interpolation of the solution on the boundary of subdomains. It is proved that the solution of the difference problem converges in uniform norm to that of differential problem with the order $O(\tau^2+h^2)$.