Optimal error estimates of a locally one-dimensional method for the multidimensional heat equation

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 $O\bigl((\tau+|h|^2)^\gamma\bigr)$, $1/2\le\gamma<1$.