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  "body": "\n## Summary\n\nThe finite element method is a way to compute approximate solutions of boundary value problems for [[partial differential equations|GC5BDZ6QBC]].\nIt writes the problem in [[weak form|SJZX81A783]] and splits the domain into small pieces such as triangles.\nThen it looks for the best approximation among functions that are polynomials on each piece.\nIt is a [[Galerkin method|Y5B0PEBKE8]] whose basis functions are supported on a few pieces, so it leads to a large sparse linear system.\nIt handles complicated geometries well and is the standard tool of engineering simulation in structural mechanics, heat flow and electromagnetism.\n\n## Explanations\n\n### L1 human\n\nCover the region with a mesh of small triangles.\nOn each triangle, approximate the unknown function by a flat (linear) piece, and glue the pieces so that the result is continuous.\nEach such function is fixed by its values at the corners of the triangles, so the problem becomes finding finitely many numbers.\nThe equation is not imposed point by point, but in an averaged sense against every function of the same kind.\nFor the equation $-u'' = f$ on an interval with \"hat\" functions on a uniform grid, this gives the tridiagonal system with rows $(-1, 2, -1)/h$.\n",
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