Sharp error estimates of some two-level methods of solving the three-dimensional heat equation

The initial-boundary value problem $\partial u/\partial t-\Delta u=f$ in $Q=\Omega\times(0,T)$, $u|_{\partial\Omega\times(0,T)}=0$, $u|_{t=0}=u_0$, is solved, where $\Omega$ is a three-dimensional rectangular parallelepiped. Two-level methods of second-order approximation are considered: families of projection and finite-difference schemes with a splitting operator as well as Crank–Nicolson schemes. Error estimates in $L_2(Q)$ of order $O(\tau^{1+\alpha}+h^2)$ for all $0\leqslant\alpha\leqslant1$ are derived. It is shown that the inclusion of values $0<\alpha\leqslant1$ yields sharpened estimates when $f$ is discontinuous. Accuracy of the estimates with respect to order – and in the case of Crank–Nicolson schemes their unimprovability – is proved. It is found that for difference schemes with splitting operator when $0<\alpha\leqslant1$, $f$ must have in $Q$ not only order $\alpha$ smoothness with respect to $t$ (as in the case of Crank–Nicolson schemes) but also order $2\alpha$ smoothness (in a certain weak sense) in the space variables. Only one scheme with splitting operator out of each family constitutes an important exception, a scheme equivalent to one proposed by J. Douglas and its projective analogue, and that only for $0<\alpha\leqslant1/2$. The situation described is qualitatively different from those studied previously in the literature.
Bibliography: 17 titles.