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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional real vector space with an inner product $a(\\cdot, \\cdot)$.\nLet $V_0, V_1, \\dots, V_N$ be subspaces of $V$ with $V = V_0 + V_1 + \\cdots + V_N$.\nLet $P_i \\colon V \\to V_i$ be the $a$-[[orthogonal projection|0B8KZRT28M]] onto $V_i$, and let $P = P_0 + P_1 + \\cdots + P_N$ be the *additive Schwarz operator*.\nThen $P$ is self-adjoint and positive definite with respect to $a$, and for every $v \\in V$\n$$a(P^{-1} v, v) = \\min \\Bigl\\{ \\sum_{i=0}^{N} a(v_i, v_i) : v_i \\in V_i, \\ \\sum_{i=0}^{N} v_i = v \\Bigr\\}.$$\nSuppose that every $v \\in V$ has a decomposition $v = \\sum_i v_i$ with $v_i \\in V_i$ and $\\sum_i a(v_i, v_i) \\le C_0^2 \\, a(v, v)$.\nThen the eigenvalues of $P$ lie in $[C_0^{-2}, N + 1]$.\nSo the [[condition number|AB3S6MHJ8X]] of $P$ is at most $(N + 1) C_0^2$.\n\n## Remarks\n\nEach $P_i$ is self-adjoint and satisfies $0 \\le a(P_i w, w) \\le a(w, w)$, so $P$ is self-adjoint with eigenvalues at most $N + 1$.\nIf $a(Pw, w) = 0$, then $P_i w = 0$ for all $i$, so $w$ is orthogonal to every $V_i$ and hence to $V$, and $w = 0$.\nFor the identity, the choice $v_i = P_i P^{-1} v$ gives $\\sum_i a(v_i, v_i) = a(v, P^{-1} v)$.\nFor any other decomposition, $a(v, P^{-1} v) = \\sum_i a(v_i, P_i P^{-1} v)$, and the Cauchy–Schwarz inequality gives $a(v, P^{-1} v) \\le \\sum_i a(v_i, v_i)$.\n\nIn matrix form, let $A$ be the symmetric positive definite stiffness matrix and let $R_i$ be the restriction to the unknowns of subdomain $i$.\nThen the method uses the [[preconditioner|KE98SW3SBD]] $M^{-1} = \\sum_i R_i^T (R_i A R_i^T)^{-1} R_i$, and $M^{-1} A$ is the matrix of $P$.\nOne applies it inside the [[conjugate gradient method|AFXZGQB4EN]], and the local solves can run in parallel.\nTake finite elements on overlapping subdomains of diameter $H$ with overlap $\\delta$, and a coarse space $V_0$ on the coarse grid.\nThen the standard bound is $\\kappa(P) \\le C (1 + H/\\delta)$, with $C$ independent of the mesh size and the number of subdomains.\nWithout the coarse space, the bound degrades as the number of subdomains grows, because a single subdomain solve cannot carry information across the whole domain.\n\n## Explanations\n\n### L1 human\n\nThe additive Schwarz method splits a big problem into many overlapping small ones, solves all of them at the same time, and adds the corrections.\nUsed alone this would overshoot, so it is used as a preconditioner: it turns a hard linear system into an easier one for an iterative solver.\nThe theorem says the result is well conditioned as long as every function can be split into local pieces whose total energy is not much more than its own.\n\n### L2 human\n\nThe identity for $a(P^{-1} v, v)$ is often called Lions' lemma, and it reduces the analysis to finding a stable splitting.\nIt is the parallel counterpart of the [[Schwarz alternating method|97VNBS15CD]], just as the Jacobi method is the parallel counterpart of Gauss–Seidel.\nColoring the subdomains so that overlapping ones get different colors improves the bound $N + 1$ to one plus the number of colors.\nThe same abstract framework covers [[multigrid methods|AC9PY6FBYF]] viewed as subspace corrections.\n",
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