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  "body": "\n## Statement\n\nLet $A_h$ be an [[invertible|0EB9N0S33C]] real $n \\times n$ matrix (the fine-grid matrix) and $A_H$ an invertible real $N \\times N$ matrix with $N \u003c n$ (the coarse-grid matrix).\nLet $P$ be an $n \\times N$ matrix (prolongation) and $R$ an $N \\times n$ matrix (restriction).\nLet $W$ be an invertible $n \\times n$ matrix, and let $S = I - W^{-1} A_h$ be the iteration matrix of the smoother $x \\mapsto x + W^{-1}(b - A_h x)$.\nOne step of the *two-grid method* for $A_h x = b$ applies $\\nu \\ge 1$ smoothing steps to $x$ and then the coarse-grid correction $x \\mapsto x + P A_H^{-1} R (b - A_h x)$.\n\n1. The error $e = x - A_h^{-1} b$ is mapped to $M e$, where $M = (I - P A_H^{-1} R A_h) S^{\\nu}$.\n2. (Hackbusch) Let $\\norm{\\cdot}$ be an operator norm, and let $C \u003e 0$ and $\\eta(\\nu) \\ge 0$ be numbers with $\\norm{A_h^{-1} - P A_H^{-1} R} \\le C / \\norm{A_h}$ (approximation property) and $\\norm{A_h S^{\\nu}} \\le \\eta(\\nu) \\norm{A_h}$ (smoothing property). Then $\\norm{M} \\le C \\, \\eta(\\nu)$.\n\n## Remarks\n\nClause 1 follows by composing the error maps $e \\mapsto S e$ of the smoother and $e \\mapsto (I - P A_H^{-1} R A_h) e$ of the correction.\nClause 2 follows from the factorization $M = (A_h^{-1} - P A_H^{-1} R) \\, A_h S^{\\nu}$ and the two bounds.\nConsider a family of [[finite difference|X7HYEBA5SA]] or finite element discretizations of an [[elliptic equation|B4TTNHSTRX]], with mesh widths $h$ and $H = 2h$.\nUse damped [[Jacobi|B8FVHNXAN1]] smoothing and standard transfer matrices.\nUnder elliptic regularity assumptions, both properties hold with $C$ and $\\eta$ independent of $h$, and $\\eta(\\nu) \\to 0$.\nThen for $\\nu$ large enough the two-grid method contracts the error by a factor less than $1$ that does not depend on $h$.\nExample: for $A_h$ symmetric positive definite and $S = I - A_h / \\norm{A_h}_2$ in the [[matrix norm|54T6Y6DSY4]] $\\norm{\\cdot}_2$, one has $\\norm{A_h S^\\nu}_2 \\le \\max_{0 \\le t \\le 1} t (1 - t)^\\nu \\norm{A_h}_2 \\le \\norm{A_h}_2 / (\\nu + 1)$.\nSolving the coarse problem recursively by the same method gives the V-cycle and the W-cycle.\n\n## Explanations\n\n### L1 human\n\nSimple methods such as Jacobi or [[Gauss–Seidel|G0EDYEDG4S]] remove the rough, oscillating part of the error quickly, but the smooth part only very slowly.\nA smooth error can be represented well on a coarser grid with far fewer unknowns, where it is cheap to remove.\nMultigrid alternates the two: smooth on the fine grid, correct on the coarse grid, and repeat over a hierarchy of grids.\n\n### L2 human\n\nFor elliptic problems, multigrid reaches a fixed accuracy in $O(n)$ operations, which is optimal up to a constant.\nThe analysis splits into the smoothing property, a statement about the smoother alone, and the approximation property, a statement about the discretizations on the two grids.\nAlgebraic multigrid builds the coarse levels from the matrix alone, without a geometric grid.\nMultigrid is also used as a [[preconditioner|KE98SW3SBD]] for the [[conjugate gradient method|AFXZGQB4EN]], and it is among the fastest [[iterative methods|2RPE77ACER]] for such problems.\n",
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