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  "body": "\n## Summary\n\nThe Galerkin method approximates the solution of a problem in weak form, $B(u, v) = f(v)$ for all $v$ in a [[Hilbert space|HM2HHE3J51]].\nIt solves the same problem on a finite-dimensional subspace.\nThis turns a partial differential equation into a finite system of linear equations.\nFor a bounded and [[coercive|86KRHS6X22]] form, the [[Lax–Milgram theorem|85KK79FCN5]] gives existence and uniqueness of the approximation.\nIts error is within a fixed factor of the best possible approximation from the subspace.\nThe finite element method and spectral methods are Galerkin methods, and Galerkin approximations are also used to prove existence of [[weak solutions|SJZX81A783]].\n\n## Explanations\n\n### L1 human\n\nThe true solution lives in an infinite-dimensional space of functions.\nThe Galerkin method picks a few simple functions, such as polynomials or piecewise linear \"hat\" functions, and looks for the best combination of them.\n\"Best\" means that the equation holds when tested against each of the chosen functions.\nThis gives as many equations as unknown coefficients, which a computer can solve.\n",
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