Title: Two-component domain decomposition scheme with overlapping subdomains for parabolic equations,
Abstract: An iteration-free method of domain decomposition is considered for
approximate solving a boundary value problem for a second-order parabolic
equation. A standard approach to constructing domain decomposition schemes is
based on a partition of unity for the domain under the consideration. Here a
new general approach is proposed for constructing domain decomposition schemes
with overlapping subdomains based on indicator functions of subdomains. The
basic peculiarity of this method is connected with a representation of the
problem operator as the sum of two operators, which are constructed for two
separate subdomains with the subtraction of the operator that is associated
with the intersection of the subdomains. There is developed a two-component
factorized scheme, which can be treated as a generalization of the standard
Alternating Direction Implicit (ADI) schemes to the case of a special
three-component splitting. There are obtained conditions for the unconditional
stability of regionally additive schemes constructed using indicator functions
of subdomains. Numerical results are presented for a model two-dimensional
problem.