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  "body": "\n## Statement\n\nLet $V$ and $Q$ be real [[Hilbert spaces|HM2HHE3J51]].\nLet $a \\colon V \\times V \\to \\R$ and $b \\colon V \\times Q \\to \\R$ be bounded [[bilinear forms|0SQZMXB6GP]], and let $f \\in V'$ and $g \\in Q'$, where $'$ denotes the [[continuous dual space|4A82S5FVRT]].\nConsider the saddle-point problem: find $(u, p) \\in V \\times Q$ with\n$$a(u, v) + b(v, p) = f(v) \\ \\text{ for all } v \\in V, \\qquad b(u, q) = g(q) \\ \\text{ for all } q \\in Q.$$\nLet $V_h \\subseteq V$ and $Q_h \\subseteq Q$ be finite-dimensional subspaces, and let $Z_h = \\set{v \\in V_h : b(v, q) = 0 \\text{ for all } q \\in Q_h}$.\nAssume that there are constants $\\alpha \u003e 0$ and $\\beta \u003e 0$, independent of $h$, such that:\n\n1. $a$ is [[coercive|86KRHS6X22]] on $Z_h$: $a(v, v) \\ge \\alpha \\norm{v}_V^2$ for all $v \\in Z_h$;\n2. the inf-sup condition, or LBB condition, holds:\n$$\\inf_{q \\in Q_h \\setminus \\set{0}} \\ \\sup_{v \\in V_h \\setminus \\set{0}} \\frac{b(v, q)}{\\norm{v}_V \\norm{q}_Q} \\ge \\beta.$$\n\nThen the discrete problem, posed with $V_h$ and $Q_h$ in place of $V$ and $Q$, has a unique solution $(u_h, p_h)$.\nIf the continuous problem has a solution $(u, p)$, then\n$$\\norm{u - u_h}_V + \\norm{p - p_h}_Q \\le C \\Big( \\inf_{v \\in V_h} \\norm{u - v}_V + \\inf_{q \\in Q_h} \\norm{p - q}_Q \\Big),$$\nwhere $C$ depends only on $\\alpha$, $\\beta$ and the bounds of $a$ and $b$.\n\n## Remarks\n\nThe same two conditions with $V$, $Q$ and the [[kernel|S417YNEJ6M]] $Z = \\set{v \\in V : b(v, q) = 0 \\text{ for all } q \\in Q}$ give existence and uniqueness for the continuous problem; this is Brezzi's theorem.\nThe inf-sup condition says that the operator $B \\colon V \\to Q'$ given by $b$ maps onto $Q'$ with a bounded right inverse.\nSo one first finds $u_g$ with $b(u_g, q) = g(q)$ and $\\norm{u_g} \\le \\norm{g} / \\beta$.\nThen one solves for $u - u_g$ in $Z$ by the [[Lax–Milgram theorem|85KK79FCN5]].\nFinally one recovers $p$ from the first equation, using the inf-sup condition again.\nThe point of the discrete theorem is that $\\beta$ does not depend on $h$; otherwise the constant $C$ can blow up as the mesh is refined.\n\nExample: for the Stokes equations, $V = H^1_0(\\Omega)^d$, $Q = L^2_0(\\Omega)$, $a(u, v) = \\int_\\Omega \\nabla u : \\nabla v$ and $b(v, q) = -\\int_\\Omega q \\, \\nabla \\cdot v$.\nThe Taylor–Hood pair, continuous piecewise quadratic velocities with continuous piecewise linear pressures, satisfies the inf-sup condition uniformly on suitable meshes.\n\nNecessity: the pair of continuous piecewise linear velocities and pressures on a triangle mesh violates the condition.\nThe discrete problem then has spurious pressure modes or a constant $\\beta_h$ that tends to $0$, and the computed pressure oscillates.\n\n## Explanations\n\n### L1 human\n\nSome problems have two unknowns that play different roles, such as the velocity and the pressure of a slow viscous flow.\nWhen both are approximated by finite elements, the two spaces must be balanced.\nIf the pressure space is too rich compared with the velocity space, the pressure is not controlled and the computation fails.\nThe inf-sup condition is the exact test of this balance.\n\n### L2 human\n\nThe condition is the key stability requirement of [[mixed finite element methods|87VMQ892VW]].\nIt replaces the coercivity of the Lax–Milgram setting, which a saddle-point problem never has on the whole space $V \\times Q$.\nChecking it for a given pair of spaces is often done with a Fortin operator: a bounded projection $\\Pi_h \\colon V \\to V_h$ with $b(v - \\Pi_h v, q) = 0$ for all $q \\in Q_h$.\nStable pairs include Taylor–Hood elements for Stokes flow and [[Raviart–Thomas elements|DVB79BK06Y]] with piecewise constants for the mixed Poisson problem.\n",
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