Fourier spectral method

A Fourier spectral method represents a periodic unknown by finitely many Fourier modes and converts differentiation into multiplication by wavenumber.

Use $N$ equally spaced samples on a period of length $L$, excluding the repeated endpoint. Their spacing is $L/N$, and the discrete Fourier transform angular wavenumbers are $2\pi m/L$. Including both endpoints in an $N$-point array gives spacing $L/(N-1)$ and is inconsistent with that wavenumber assignment. It does not supply an extra independent periodic sample.

A spectral truncation retains a finite set of basis modes and sets all omitted coefficients to zero.

For the two-periodic problemthe Fourier coefficients satisfy the tridiagonal system

The Fourier–Galerkin method seeks a finite Fourier series whose residual is orthogonal to every retained Fourier mode.