  The reduction of computational costs in the numerical solution of
nonstationary problems is achieved through splitting schemes. In this case,
solving a set of less computationally complex problems provides the transition
to a new level in time. The traditional construction approach of splitting
schemes is based on an additive representation of the problem operator(s) and
uses explicit-implicit approximations for individual terms. Recently (Y.
Efendiev, P.N. Vabishchevich. Splitting methods for solution decomposition in
nonstationary problems. \textit{Applied Mathematics and Computation}.
\textbf{397}, 125785, 2021), a new class of methods of approximate solution of
nonstationary problems has been introduced based on decomposition not of
operators but of the solution itself. This new approach with subdomain solution
selection is used in this paper to construct domain decomposition schemes. The
boundary value problem for a second-order parabolic equation in a rectangle
with a difference approximation in space is typical. Two and three-level
schemes for decomposition of the domain with and without overlapping subdomains
are investigated. Our numerical experiments complement the theoretical results.
