On difference schemes on grids locally refining in time (Q1909823)

The author discusses stable and accurate finite difference schemes for parabolic problems. The proposed algorithms are based on the domain decomposition method. The domain is divided into a finite number of subdomains and the computational grids (in space and time) are chosen independently in each subdomain. This includes also the case when over each subdomain the time steps may differ. The boundary conditions on the subdomain interfaces are treated using appropriate interpolation. The proposed approach generalizes some previous results of \textit{P. P. Matus} [Differ. Uravn. 27, No. 11, 1964-1974 (1991; Zbl 0743.65075)] and of \textit{R. E. Ewing}, \textit{R. D. Lazarov} and \textit{P. S. Vassilevski} [Computing 45, No. 3, 193-215 (1990; Zbl 0721.65047)] in the following sense: the introduced subdomain overlap allows to control the error of the method. For generous overlap there is no loss of accuracy compared with the standard schemes; for schemes without overlap the developed theory produces some previously known results.