Geodesic
Crank--Nicolson's scheme with different time-step in subdomains for the solution of parabolic problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 2, pp. 137-150

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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 $O(\tau^2+h^2)$. 
@article{SJVM_2001_4_2_a2,
     author = {V. I. Drobyshevich and L. N. Katkova},
     title = {Crank--Nicolson's scheme with different time-step in subdomains for the solution of parabolic problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {137--150},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_2_a2/}
}
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V. I. Drobyshevich; L. N. Katkova. Crank--Nicolson's scheme with different time-step in subdomains for the solution of parabolic problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 2, pp. 137-150. http://geodesic.mathdoc.fr/item/SJVM_2001_4_2_a2/