Geodesic
Optimal error estimates of a~locally one-dimensional method for the multidimensional heat equation
Matematičeskie zametki, Tome 60 (1996) no. 2, pp. 185-197

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For the multidimensional heat equation in a parallelepiped, optimal error estimates in $L_2(Q)$ are derived. The error is of the order of $O(\tau+|h|^2)$ for any right-hand side $f\in L_2(Q)$ and any initial function $u_0\in\mathring W_2^1(\Omega)$; for appropriate classes of less regular $f$ and $u_0$, the error is of the order of $O\bigl((\tau+|h|^2)^\gamma\bigr)$, $1/2\le\gamma1$. 
@article{MZM_1996_60_2_a2,
     author = {S. B. Zaitseva and A. A. Zlotnik},
     title = {Optimal error estimates of a~locally one-dimensional method for the multidimensional heat equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {185--197},
     publisher = {mathdoc},
     volume = {60},
     number = {2},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_2_a2/}
}
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S. B. Zaitseva; A. A. Zlotnik. Optimal error estimates of a~locally one-dimensional method for the multidimensional heat equation. Matematičeskie zametki, Tome 60 (1996) no. 2, pp. 185-197. http://geodesic.mathdoc.fr/item/MZM_1996_60_2_a2/