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  "body": "\n## Summary\n\nFinite element exterior calculus is a framework for designing and analysing finite element methods with the tools of [[differential forms|0E3G0GRPZ6]] and [[de Rham cohomology|GX74C07T0G]].\nIt was developed by Douglas Arnold, Richard Falk and Ragnar Winther.\nIts central idea is that finite element spaces should form a subcomplex of the de Rham complex, with projections that commute with the [[exterior derivative|0ZHSTAG8ZY]].\nSuch spaces give stable [[mixed finite element methods|87VMQ892VW]] for the Hodge Laplacian and explain many earlier elements in one theory.\nSee the [[definition|GRBZT7SWK0]].\n\n## Explanations\n\n### L1 human\n\nIn three dimensions, the gradient, the curl and the divergence form a chain: the curl of a gradient is zero, and so is the divergence of a curl.\nGood numerical methods for electromagnetism or fluid flow must respect this chain.\nFinite element exterior calculus builds matching chains of finite element spaces, one space for each link.\nFor example, continuous piecewise linear functions, Nédélec edge elements, Raviart–Thomas face elements and piecewise constants form such a chain.\n",
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