Domain decomposition methods with overlapping subdomains for time-dependent problems

TL;DR

This paper reviews overlapping domain decomposition methods for time-dependent problems, focusing on operator-splitting schemes and data exchange strategies that ensure convergence across different function spaces.

Contribution

It provides a comprehensive classification of DD methods, highlighting the importance of overlapping subdomains and operator-splitting schemes for stable numerical solutions.

Findings

Overlapping subdomain methods are preferable for homogeneous algorithms.

Different data exchange strategies impact convergence properties.

Operator-splitting schemes are crucial for solving time-dependent problems.

Abstract

Domain decomposition (DD) methods for solving time-dependent problems can be classified by (i) the method of domain decomposition used, (ii) the choice of decomposition operators (exchange of boundary conditions), and (iii) the splitting scheme employed. To construct homogeneous numerical algorithms, overlapping subdomain methods are preferable. Domain decomposition is associated with the corresponding additive representation of the problem operator. To solve time-dependent problems with the DD splitting, different operator-splitting schemes are used. Various variants of decomposition operators differ by distinct types of data exchanges on interfaces. They ensure the convergence of the approximate solution in various spaces of grid functions.