Iterative Refinement Loops: Exploring Unconventional Validation in Spectral Methods
Why Validation Loops Matter Now for Spectral Practitioners Spectral methods are the gold standard for problems with smooth solutions—exponential convergence in theory, but brittleness in practice. A single unresolved boundary layer or a slight mismatch in boundary conditions can produce solutions that look plausible yet are completely wrong. Traditional validation approaches—method of manufactured solutions (MMS), grid refinement studies, and residual checks—work well for many discretizations, but they have blind spots. MMS requires an analytic source term that may not exercise all nonlinear interactions in the code. Grid refinement assumes the error decays monotonically, which spectral schemes often violate due to the Gibbs phenomenon or aliasing. Residual checks can pass even when the solution is contaminated by spurious eigenvalues.