Validation — trusting the substrate

Before any adaptive machinery enters, two questions must be settled.

(A) Does the finite-element method converge? — does the discrete eigenvalue \lambda_h approach the true \lambda as the mesh refines, at the rate the theory predicts? This is a question about the discretisation, independent of any parameter sweep.

(B) Is the fast assembly faithful? — the stiffness matrix K(\mu) = \sum_{ij} c_{ij}(\mu)\, K_{ij} is built cheaply by recombining four precomputed component matrices. Does this match what direct finite-element assembly would give, to floating-point round-off?

Both building blocks are shared by the surrogate and the dense reference used to validate it. A flaw in either is invisible to any downstream equivalence check, because both sides carry the same error. These two pages answer the questions independently, before the algorithm is invoked.

Pages

Spectrum samples

The raw finite-element assembly — no surrogate, no parameter sweep — is checked against the closed-form eigenvalues (n\pi)^2 of the Laplacian on [0,1]^2. The page separates two distinct claims: a visual wiring smoke test (is the assembly wired correctly?) and a falsifiable convergence rate study (does \lambda_h \to \lambda at the expected O(h^2) rate?).

Affine-cache cross-check

The diffusion coefficient c(\mu) is affine in \mu, so the stiffness matrix decomposes into four fixed component matrices assembled once and recombined per parameter value. This page verifies that the recombined assembly reproduces direct finite-element assembly to floating-point round-off — so the speed-up costs nothing in accuracy.