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  "body": "\n## Definition\n\nLet $\\Omega \\subset \\R^n$ be a bounded Lipschitz polyhedral domain.\nFor $0 \\le k \\le n$, let $H\\Lambda^k(\\Omega)$ be the [[differential $k$-forms|0E3G0GRPZ6]] $\\omega$ with $\\omega \\in L^2$ and $d\\omega \\in L^2$.\nHere $d$ is the [[exterior derivative|0ZHSTAG8ZY]] in the weak sense.\nThese [[Hilbert spaces|HM2HHE3J51]] form the $L^2$ de Rham complex $H\\Lambda^0 \\xrightarrow{d} H\\Lambda^1 \\xrightarrow{d} \\cdots \\xrightarrow{d} H\\Lambda^n$.\n*Finite element exterior calculus* discretizes this complex by a family of spaces $V_h^k \\subseteq H\\Lambda^k(\\Omega)$, indexed by a mesh parameter $h$, with three properties:\n\n1. each $V_h^k$ is a [[finite element|98GZ9DVHTA]] space of piecewise polynomial $k$-forms on a simplicial mesh of $\\Omega$;\n2. subcomplex: $d V_h^k \\subseteq V_h^{k+1}$ for each $k$;\n3. bounded cochain projections: there are projections $\\pi_h^k$ onto $V_h^k$ with $d \\pi_h^k = \\pi_h^{k+1} d$.\n\nThe projections in clause 3 are required to be bounded in $L^2$ uniformly in $h$.\nThe method is the [[mixed|87VMQ892VW]] Galerkin method for the [[Hodge Laplacian|PWG3GKZG80]] on these spaces.\n\n## Remarks\n\nThe standard families are $\\mathcal{P}_r \\Lambda^k$ and $\\mathcal{P}_r^- \\Lambda^k$ of polynomial $k$-forms.\nExample: in $\\R^3$, take continuous piecewise linear functions, lowest-order Nédélec edge elements, lowest-order Raviart–Thomas face elements, and piecewise constants.\nThe gradient of a piecewise linear function is piecewise constant with continuous tangential part, so it lies in the Nédélec space.\nThe other inclusions hold for the same reason, and the canonical interpolation operators commute with $d$.\nUnder clauses 2 and 3 the discrete complex has the same [[cohomology|GX74C07T0G]] as the continuous one, and discrete Poincaré inequalities hold uniformly in $h$.\n\nNon-example: take continuous piecewise quadratics for $V_h^0$ and lowest-order Nédélec edge elements for $V_h^1$.\nThe lowest-order Nédélec fields on a tetrahedron have the form $a + b \\times x$.\nThe gradient of a quadratic such as $x_1^2$ is $(2 x_1, 0, 0)$, which is not of this form, so clause 2 fails.\n",
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