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Gibbs
Discontinuities create persistent oscillations that spectral accuracy cannot remove without special treatment.
Chebyshev
Chebyshev nodes cluster near endpoints and are effective for finite intervals.
Aliasing
Nonlinear terms create unresolved modes; dealiasing or filtering may be needed.
Differentiation matrix
Pseudospectral differentiation matrices map nodal values to derivative approximations.
Fourier
Fourier methods fit periodic problems and transform differentiation into multiplication in mode space.
Spectral approximation
Spectral methods use global polynomial or trigonometric bases and can converge extremely fast for smooth functions.