Spectral methods use global basis functions such as Fourier or Chebyshev modes. For smooth solutions, they can converge much faster than low-order local methods.
Spectral approximation
Definition. A spectral method approximates a function by a truncated expansion in global basis functions, typically Fourier modes for periodic domains or Chebyshev polynomials on intervals.
Spectral convergence
Theorem. For analytic functions, coefficients in suitable spectral expansions decay exponentially. Truncation error can therefore decay faster than any algebraic power of the resolution parameter.
Pseudospectral collocation
Definition. A pseudospectral method represents the solution globally but enforces equations at collocation nodes. Fast transforms make many Fourier and Chebyshev implementations efficient.
Aliasing
Proposition. Nonlinear products generate frequencies beyond the truncated basis. Without dealiasing, these unresolved modes fold back into lower modes and contaminate the computation.
Smoothness boundary
Remark. Spectral methods are excellent for smooth solutions on simple domains. Discontinuities, shocks, complex geometry, and local singularities often favor finite volume, finite difference, or finite element methods.
Global approximation
Spectral methods represent solutions by global trigonometric or orthogonal polynomial bases. For periodic \(u\), \[ u_N(x)=\sum_{|k|\le N}\widehat u_ke^{ikx}. \] Differentiation is diagonal: \[ \widehat{u_N'}_k=ik\widehat u_k. \] The FFT computes transforms in \(O(N\log N)\).
Convergence from regularity
Theorem. If periodic \(u\in H^s\), then its Fourier projection satisfies \[ \|u-P_Nu\|_{H^r} \le N^{r-s}\|u\|_{H^s}, \qquad 0\le r\le s. \] If \(u\) extends analytically to a strip \(|\operatorname{Im}z|<\alpha\), then Fourier coefficients and projection error decay exponentially like \(e^{-\alpha N}\).
Proof. Parseval gives \[ \|u-P_Nu\|_{H^r}^2 =\sum_{|k|>N}(1+k^2)^r|\widehat u_k|^2 \le(1+N^2)^{r-s}\|u\|_{H^s}^2. \] For analytic \(u\), shift the contour in the Fourier coefficient integral by \(\pm i\alpha'\), \(\alpha'<\alpha\), obtaining \(|\widehat u_k|\le Ce^{-\alpha'|k|}\).
Chebyshev collocation
On \([-1,1]\), Chebyshev--Lobatto nodes \[ x_j=\cos(j\pi/N) \] cluster near endpoints. Barycentric interpolation yields a dense differentiation matrix \(D_N\), so derivatives at nodes are \(D_Nu\). Fast cosine transforms avoid dense multiplication for repeated operations. Boundary conditions can be imposed by row replacement, basis construction, or tau methods.
Aliasing and dealiasing
Products of two degree-\(N\) expansions contain modes through \(2N\). On an \(N\)-point grid, unresolved modes fold into lower frequencies. The \(3/2\) rule pads Fourier coefficients, transforms to a finer grid, multiplies, then truncates. Without dealiasing, nonlinear conservation and stability can be corrupted.
Gibbs phenomenon
At a jump, spectral partial sums overshoot by a nonvanishing fraction of the jump, although convergence holds away from it and in \(L_2\). Filtering, domain decomposition, Gegenbauer reconstruction, or shock-capturing can reduce oscillation. Global spectral methods are strongest for smooth solutions; they do not turn discontinuities into exponential convergence.