Two-component domain decomposition scheme with overlapping subdomains for parabolic equations

An iteration-free method of domain decomposition is considered for approximately solving a boundary value problem for a second-order parabolic equation. A standard approach for 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. The present paper proposed 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. The scheme is regionally additive and is constructed using indicator functions of the subdomains. Moreover, it is unconditionally stable if the weight is chosen to be greater than or equal to 0.5. Numerical results are presented for a model two-dimensional problem.