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  "body": "\n## Summary\n\nA spectral method approximates the solution of a differential equation by a finite sum of smooth global functions, such as trigonometric functions or [[Chebyshev polynomials|M6EAR1JHPP]].\nThe coefficients are fixed by a [[Galerkin|Y5B0PEBKE8]] condition, a tau condition or collocation at chosen points.\nFor smooth solutions the error decreases faster than any power of the number of terms, often exponentially.\nSo they are the method of choice for smooth problems on simple domains.\nExamples are turbulence in periodic boxes and weather models on the sphere.\nSee the [[definition|QFJEFGR171]].\n\n## Explanations\n\n### L1 human\n\nA finite element method builds the solution from many small local pieces.\nA spectral method instead uses a few waves that each stretch over the whole domain, as in a [[Fourier series|G12KXDHPPS]].\nDerivatives of waves are easy: differentiating $e^{ikx}$ just multiplies it by $ik$.\nFor a smooth solution, the wave amplitudes drop very fast, so a few dozen waves can give many correct digits.\n",
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