Spectral methods use global polynomial expansions on Chebyshev–Gauss–Lobatto points x_j = cos(πj/N). Unlike finite differences, they achieve exponential ("spectral") convergence for smooth functions. Visualize the approximation error vs. polynomial degree N.
Chebyshev nodes cluster at endpoints, preventing Runge's phenomenon. For analytic functions: error ∝ ρ^(-N) where ρ > 1 (geometric convergence). For C^k functions: O(N^(-k)). Singular functions like |x|^1.5 show algebraic convergence only.