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

## TL;DR

This paper introduces a novel domain decomposition method for parabolic equations using overlapping subdomains and indicator functions, generalizing ADI schemes with proven unconditional stability and demonstrated through numerical experiments.

## Contribution

It proposes a new overlapping subdomain scheme based on indicator functions and a three-component splitting, extending ADI methods with stability analysis.

## Key findings

- Unconditional stability conditions established for the new schemes.
- Numerical results confirm the effectiveness of the method.
- Generalization of ADI schemes to three-component splitting.

## 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.