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  "body": "\n## Description\n\nLet $T \\subset \\R^d$ be a simplex, a triangle when $d = 2$, and let $k \\ge 0$ be an integer.\nLet $P_k(T)$ be the [[polynomials|07GNX8GCYP]] of degree at most $k$ on $T$, and $\\tilde P_k(T)$ the homogeneous ones of degree $k$.\nThe *Raviart–Thomas space* of index $k$ on $T$ is\n$$RT_k(T) = P_k(T)^d + x \\, \\tilde P_k(T),$$\nthe vector fields $p(x) + x \\, q(x)$ with $p \\in P_k(T)^d$ and $q \\in \\tilde P_k(T)$.\nThe degrees of freedom are the moments of the normal component $v \\cdot n$ on each face against $P_k$ of that face.\nFor $k \\ge 1$ they also include the moments of $v$ against $P_{k-1}(T)^d$.\nOn a mesh, the global space is formed by the piecewise $RT_k$ fields whose normal moments agree on each interior face.\n\n## Remarks\n\nLowest order on a triangle: $RT_0(T) = \\set{x \\mapsto a + b x : a \\in \\R^2, \\ b \\in \\R}$, of dimension $3$.\nOn an edge $e$ with unit normal $n$, the points satisfy $x \\cdot n = c$ for a constant $c$.\nSo $(a + b x) \\cdot n = a \\cdot n + b c$ is constant on $e$, and it is fixed by the flux $\\int_e v \\cdot n$.\nThe three fluxes determine $v$, which gives the three degrees of freedom.\n\nThe dimension of $RT_k(T)$ on a triangle is $(k + 1)(k + 3)$.\nMatching the normal moments makes the normal component continuous across faces.\nSo the global space lies in $H(\\operatorname{div})$, the $L^2$ fields with [[divergence|0MWV6ZTK6S]] in $L^2$.\nThe divergence maps $RT_k(T)$ onto $P_k(T)$.\nPaired with discontinuous piecewise polynomials of degree $k$, these spaces give a stable [[mixed finite element method|87VMQ892VW]] for the Poisson equation.\nThey were introduced by Pierre-Arnaud Raviart and Jean-Marie Thomas.\n\n## Explanations\n\n### L1 human\n\nRaviart–Thomas elements are building blocks for approximating flows, such as the flux of heat or of water through soil.\nInstead of storing values at the corners, the lowest-order element stores how much flows through each edge of a triangle.\nNeighbouring triangles share these edge fluxes, so whatever leaves one triangle enters the next.\nThis makes the computed flow conserve mass exactly, element by element.\n",
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