Editing Spectral method (section)

== A relationship with the spectral element method ==

One can show that if <math>g</math> is infinitely differentiable, then the numerical algorithm using Fast Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a <math>C_n<\infty</math> such that the error is less than <math>C_nh^n</math> for all sufficiently small values of <math>h</math>. We say that the spectral method is of order <math>n</math>, for every n>0.

Because a [[spectral element method]] is a [[finite element method]] of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary [[elliptic boundary value problem]]s.