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  "body": "\n## Statement\n\nLet $V$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\\norm{\\cdot}$.\nLet $a \\colon V \\times V \\to \\R$ be a [[bilinear form|0SQZMXB6GP]] that is bounded, $\\abs{a(u, v)} \\le M \\norm{u} \\norm{v}$ for all $u, v \\in V$, and [[coercive|86KRHS6X22]], $a(v, v) \\ge \\alpha \\norm{v}^2$ for all $v \\in V$, with constants $M \\ge 0$ and $\\alpha \u003e 0$.\nLet $f$ be an element of the [[continuous dual space|4A82S5FVRT]] of $V$, and let $u \\in V$ satisfy $a(u, v) = f(v)$ for all $v \\in V$.\nLet $V_h$ be a closed subspace of $V$, and let $u_h \\in V_h$ satisfy $a(u_h, v) = f(v)$ for all $v \\in V_h$.\nThen\n$$\\norm{u - u_h} \\le \\frac{M}{\\alpha} \\, \\inf_{v \\in V_h} \\norm{u - v}.$$\n\n## Remarks\n\nBoth $u$ and $u_h$ exist and are unique by the [[Lax–Milgram theorem|85KK79FCN5]], applied on $V$ and on $V_h$.\nSubtracting the two equations gives Galerkin orthogonality: $a(u - u_h, w) = 0$ for all $w \\in V_h$.\nFor $v \\in V_h$, take $w = v - u_h$: then $\\alpha \\norm{u - u_h}^2 \\le a(u - u_h, u - u_h) = a(u - u_h, u - v) \\le M \\norm{u - u_h} \\norm{u - v}$.\nDividing by $\\alpha \\norm{u - u_h}$ and taking the infimum over $v$ gives the claim.\nIf $a$ is also symmetric, then $u_h$ is the $a$-orthogonal projection of $u$ onto $V_h$, and the constant improves to $\\sqrt{M / \\alpha}$.\n\nExample: for $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$, take $V = H^1_0(0, 1)$, $a(u, v) = \\int_0^1 u' v' \\, dx$, and $V_h$ the continuous piecewise linear functions on a mesh of width $h$.\nChoosing $v$ to be the piecewise linear interpolant of $u$ in the infimum gives $\\norm{u - u_h}_{H^1} \\le C h \\norm{u''}_{L^2}$ when $u \\in H^2(0, 1)$.\n\nCoercivity is needed.\nTake an indefinite form, such as $a(u, v) = \\int_0^1 (u' v' - k^2 u v) \\, dx$ with $k$ large.\nThen a discrete solution may fail to exist on a given $V_h$.\nWhen it exists, it need not be close to the best approximation unless $h$ is small enough.\n\n## Explanations\n\n### L1 human\n\nThe finite element method looks for an approximate solution inside a smaller space of simple functions.\nCéa's lemma says that this approximate solution is almost as good as the best possible approximation from that space.\nThe loss is at most a fixed factor $M / \\alpha$, which depends on the problem but not on the space.\n\n### L2 human\n\nThe lemma splits error analysis of [[Galerkin methods|Y5B0PEBKE8]] into two independent questions.\nOne is stability, measured by $M / \\alpha$, which comes from the continuous problem.\nThe other is approximation: how well functions in $V_h$ approximate the exact solution, which is a question of interpolation theory and is answered with the [[Bramble–Hilbert lemma|K4P0R83G1Q]].\nFor problems that are not coercive, such as saddle-point problems, the role of $\\alpha$ is taken by an inf-sup constant, as in the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].\n",
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