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  "body": "\n## Statement\n\nLet $V$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $a(\\cdot, \\cdot)$ and norm $\\norm{v}_a = a(v, v)^{1/2}$.\nLet $\\ell$ be a bounded linear functional on $V$, and let $u \\in V$ be the solution of $a(u, v) = \\ell(v)$ for all $v \\in V$.\nLet $V_1, V_2$ be closed subspaces of $V$ with $V_1 + V_2 = V$.\nThe *alternating Schwarz method* starts from $u^0 \\in V$ and repeats two half steps.\n\n1. Find $w_1 \\in V_1$ with $a(w_1, v) = \\ell(v) - a(u^n, v)$ for all $v \\in V_1$, and set $u^{n+1/2} = u^n + w_1$.\n2. Find $w_2 \\in V_2$ with $a(w_2, v) = \\ell(v) - a(u^{n+1/2}, v)$ for all $v \\in V_2$, and set $u^{n+1} = u^{n+1/2} + w_2$.\n\nThen there is a constant $\\rho \u003c 1$, depending only on $V_1$ and $V_2$, such that\n$$\\norm{u^n - u}_a \\le \\rho^n \\norm{u^0 - u}_a \\quad \\text{for all } n \\ge 0.$$\n\n## Remarks\n\nLet $P_i$ be the $a$-[[orthogonal projection|0B8KZRT28M]] onto $V_i$.\nThe errors satisfy $u^{n+1} - u = (I - P_2)(I - P_1)(u^n - u)$, a product of the projections onto $V_1^\\perp$ and $V_2^\\perp$.\nSince $V_1 + V_2 = V$ is closed, $V_1^\\perp \\cap V_2^\\perp = \\set{0}$ and the angle between $V_1^\\perp$ and $V_2^\\perp$ is positive, which gives $\\rho = \\norm{(I - P_2)(I - P_1)}_a \u003c 1$.\nIf $V_1 + V_2$ is only dense in $V$, von Neumann's theorem on alternating projections still gives $u^n \\to u$, but possibly without a geometric rate.\n\nThe original setting is the Dirichlet problem $-\\Delta u = f$ in $\\Omega$, $u = 0$ on $\\partial \\Omega$, where $\\Omega = \\Omega_1 \\cup \\Omega_2$ is a union of two overlapping domains.\nTake $V = H^1_0(\\Omega)$ (a [[Sobolev space|M4CEZ5K9HP]]), $a(u, v) = \\int_\\Omega \\nabla u \\cdot \\nabla v \\, dx$, and $V_i = H^1_0(\\Omega_i)$ extended by zero.\nThen each half step solves the Dirichlet problem on $\\Omega_i$ with boundary values taken from the current iterate on $\\partial \\Omega_i \\cap \\Omega$.\nA partition of unity subordinate to the overlap shows $V_1 + V_2 = V$ when the overlap has positive width.\nThe rate $\\rho$ tends to $1$ as the overlap shrinks.\n\n## Explanations\n\n### L1 human\n\nHermann Schwarz wanted to solve a boundary value problem on a region made of two overlapping simple pieces, such as a disc and a rectangle.\nHe solved on the disc, used that answer as boundary data on the part of the rectangle's edge inside the disc, solved on the rectangle, and went back and forth.\nThe theorem says this back-and-forth converges to the solution on the whole region, and the error shrinks by a fixed factor each round.\n\n### L2 human\n\nThe method is the prototype of [[domain decomposition methods|AN054RHT1K]]: a large problem is split into problems on subdomains that are cheaper to solve.\nIn the abstract form it is a block Gauss–Seidel iteration on the splitting $V = V_1 + V_2$, and the error operator is a product of orthogonal projections.\nThe [[additive Schwarz method|X68MFYW014]] is the Jacobi-type variant, which solves on all pieces at once and is used as a preconditioner.\nThe convergence rate depends on the overlap, which is why practical methods add a coarse space.\n",
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