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  "body": "\n## Definition\n\nLet $A x = b$ be a [[linear system|0GT4WWGWH2]] with an invertible real $n \\times n$ matrix $A$, arising from a discretized [[boundary value problem|TM8VQJWGHZ]] on a domain $\\Omega$.\nLet $\\Omega_1, \\dots, \\Omega_N$ 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 $\\Omega_i$.\nLet $R_i$ 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.\nA *domain decomposition method* is an [[iterative method|2RPE77ACER]] or [[preconditioner|KE98SW3SBD]] for $A x = b$ in which:\n\n1. each step solves local systems with the matrices $A_i$, or with local matrices built from them;\n2. the local solutions are combined through the unknowns shared by, or lying on the interfaces between, the subdomains;\n3. optionally, a coarse problem of small size couples all subdomains globally.\n\nThe method is *overlapping* if the sets $I_i$ overlap, and *nonoverlapping* if the subdomains meet only along interfaces.\n\n## Remarks\n\nExample: the additive Schwarz preconditioner is $M^{-1} = \\sum_{i=1}^{N} R_i^T A_i^{-1} R_i$, for overlapping $I_i$ that cover all unknowns.\nEach term solves a local problem, and the terms can be computed in parallel, so clauses 1 and 2 hold.\nIf $A$ is symmetric [[positive definite|0KX66CZMBB]], each $A_i$ is too, and $M^{-1}$ is symmetric positive definite.\nNon-example: the [[finite element method|98GZ9DVHTA]] with a direct solver for the whole of $A$ solves no local systems, so it fails clause 1.\nNonoverlapping methods eliminate the interior unknowns of each subdomain and iterate on the [[Schur complement|6HWFBXYPZP]] for the interface unknowns.\nWithout a coarse problem, the number of iterations grows with the number of subdomains, because information travels only between neighbours at each step.\n",
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