Domain decomposition methods

Definition

Let $A x = b$ be a linear system with an invertible real $n \times n$ matrix $A$, arising from a discretized boundary value problem on a domain $\Omega$. Let $\Omega_1, \dots, \Omega_N$ be subdomains with $\Omega = \Omega_1 \cup \cdots \cup \Omega_N$, and let $I_i \subseteq \set{1, \dots, n}$ be the set of unknowns that belong to $\Omega_i$. Let $R_i$ be the $\abs{I_i} \times n$ matrix that selects the entries in $I_i$, and let $A_i = R_i A R_i^T$ be the local matrix. A domain decomposition method is an iterative method or preconditioner for $A x = b$ in which:

1. each step solves local systems with the matrices $A_i$, or with local matrices built from them;
2. the local solutions are combined through the unknowns shared by, or lying on the interfaces between, the subdomains;
3. optionally, a coarse problem of small size couples all subdomains globally.

The method is overlapping if the sets $I_i$ overlap, and nonoverlapping if the subdomains meet only along interfaces.