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I don't understand what is spectral convergence. The definitions I found in google are very physics. Is there any definition in mathematics ?

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This is quoted from an online article:

Spectral convergence means that the error with increasing resolution (number of grid points $N$) is in fact decreasing exponentially, $\propto (L/N)^N$, as opposed to algebraically, $\propto (L/N)^p$ as for finite-difference methods.

[ I remember finding a description in some textbook in the library, will update when I get it next time ]

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Spectral convergence is the accuracy of a spectral method, often called spectral accuracy or convergence of infinite order.

Indeed, increasing the number, $N$, of discretization points, a spectral method presents a converge rate of order $\mathcal{O}(N^{-m})$ for every natural number $m$, provided that the solution that we want to approximate is smooth (infinitely differentiable). Moreover, if the solution is suitably analytic, it is possible to achieve even faster convergence rate, $\mathcal{O}(\mathrm{e}^{-cN})$ with $c$ positive real number, this convergence rate is also known as geometric convergence rate.

Refer to

L.N. Trefethen, Spectral Methods in Matlab (2001) https://www.mobt3ath.com/uplode/book/book-50111.pdf

Chapter 21 of L.N. Trefethen, Approximation theory and approximation practice (2011) http://www.chebfun.org/ATAP/

Further information on convergence rates (geometric convergence) can be found in Section 2.3 of J.P. Boyd, Chebyshev and Fourier Spectral Methods (2000) http://depts.washington.edu/ph506/Boyd.pdf

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