Is the finite-element substrate trustworthy?

Before any adaptive machinery enters the picture, the question is whether the finite-element discretisation the whole method rests on can be trusted. That question has two separable parts, and this page is careful to say which one each figure answers:

(A) Does the FE method converge? — does the discrete eigenvalue \lambda_h approach the true \lambda as the mesh is refined, at the rate the theory predicts? This is a statement about the discretisation.
(B) Is the assembly wired correctly? — assembly, Dirichlet BCs, and the parametric coefficient with no bug. This is a statement about the implementation.

The two eyeball figures below answer (B) — they are a wiring smoke test, not a convergence proof. The convergence section answers (A) with a falsifiable rate study, and the motivating-example overlay ties both to a first correctness check against a closed-form spectrum. Mesh-independence of the reduced model is a separate, later claim.

What these curves do and don’t show

Two axes are in play and the page keeps them apart:

The space axis is discrete. \lambda_h is an approximation — it sits near, not on, the analytic value, and the gap only closes as h \to 0 (quantified in the convergence section). A single-mesh figure landing close to (n\pi)^2 shows the plumbing is sane; it does not show the method converges.
The parameter axis is genuinely continuous. Each \lambda_n(\mu) is a real, smooth curve of the parametric operator — solved at sample points here, but a continuous object. Smoothness, stable ordering, and the absence of spurious kinks are meaningful properties of the parametric problem, and they are drawn correctly.

1D parametric Laplacian (test fixture)

Test fixture — not a physics target. Fast 1D scalar problem used to unit-test the parameter-dimension-agnostic infrastructure cheaply. Do not draw physics conclusions from it.

Model

On \Omega = (0, 1) with homogeneous Dirichlet boundary conditions,

-\bigl(a(\mu)\,u'\bigr)' \;=\; \lambda\,u, \qquad u(0)=u(1)=0,

with a piecewise-constant parametric coefficient a(\mu).

Correctness signature (wiring smoke test)

At \mu = 0, the coefficient is a \equiv 1 and the leading eigenvalues sit close to the analytic spectrum \lambda_n = (n\pi)^2 (dashed reference lines) — close, because the space axis is discrete (see the note above).
Each \lambda_n(\mu) is a smooth curve — no kinks or jumps.
Eigenvalue ordering is stable along the parameter path.

Eigenvalue-vs-µ for the parametric 1D Laplacian. Dashed lines: analytic (nπ)² at µ=0. The curves land *near* the references — a sane-plumbing check, not a convergence proof.

1D Laplacian, two-parameter coefficient

The same fixture exercised with a two-parameter coefficient: each panel sweeps one parameter axis with the other held at 0. This is the same 1D Laplacian problem — it is not the 2D diffusion model — and its purpose is to smoke-test that the parameter-dimension-agnostic infrastructure (snapshot store, sweep driver, grid) behaves identically as the parameter dimension grows. It precedes the true one-parameter benchmark on the diffusion-3-3-2 page.

Eigenvalue-vs-µ for the 1D Laplacian with a two-parameter coefficient (one axis per panel, other axis = 0).

Does the method converge?

The figures above are single-mesh snapshots; on their own they say nothing about claim (A). The convergence study does. For the 1D Laplacian at \mu=0 the exact leading eigenvalue is \lambda_1 = \pi^2, so we can refine the mesh (n_{\text{el}} \in \{25, 50, 100, 200, 400\}) and watch the discretisation error |\lambda_{1,h} - \pi^2| shrink.

Log–log mesh convergence of the leading eigenvalue. The data lie on a clean slope-2 line, and every point is *above* the exact value.

Two theory-backed properties hold, and either failing would be a real bug:

Rate. The error decays at O(h^2) — the fitted log–log slope is \approx 2, exactly the Babuška–Osborn rate for P_1 eigenvalue approximation (O(h^{2k}) for P_k). This is a falsifiable statement the single-mesh (n\pi)^2 eyeball cannot make.
From above. Every \lambda_{1,h} \ge \lambda_1. Conforming Galerkin on a subspace V_h \subset V can only overestimate eigenvalues (min–max over a smaller space), so monotone convergence from above is the signature of a correctly conforming discretisation. A point dipping below \pi^2 would signal a non-conforming bug.

The rate and the from-above property are both pinned by regression so that a regression in the assembly path is caught rather than silently degrading this page.

A motivating example

A natural first correctness check is an exact-vs-FEM overlay: a model whose spectrum is known in closed form, plotted against the computed FEM spectrum across the whole parameter range.

The model is -\operatorname{div}(A(\mu)\nabla u) = \lambda u with A(\mu) = \operatorname{diag}(1, 1+\mu) on the domain [0,1]^2 with Dirichlet BCs. The eigenfunctions are \sin(m\pi x)\sin(n\pi y) and the spectrum is closed-form:

\lambda_{n,m}(\mu) \;=\; \pi^2\bigl(m^2 + (1+\mu)\,n^2\bigr).

Exact (lines) vs FEM (dots) for the motivating example, A(µ)=diag(1,1+µ), across the full µ-range. FEM tracks the closed form from above; the curve crossings are what later motivate eigenvalue tracking.

The FEM dots track the closed-form lines across the entire \mu-range, sitting just above them (claim (A) again, now on the real 2D physics target rather than the 1D fixture). Two things to notice:

The eigenfunctions are \mu-independent here (a deliberate simplification of the motivating example), so the only \mu-dependence is in the eigenvalues — yet the curves still cross. Those crossings are exactly what motivates eigenvalue tracking and the reduced-basis rank argument, and they feed directly into the parametric sweep problems built on this substrate.
This overlay is the honest version of “trust the substrate”: a model where we know the answer, checked against the code across the whole parameter domain — not a single-mesh glance.

--- title: "Is the finite-element substrate trustworthy?" --- Before any adaptive machinery enters the picture, the question is whether the finite-element discretisation the whole method rests on can be trusted. That question has two separable parts, and this page is careful to say which one each figure answers: - **(A) Does the FE *method* converge?** — does the discrete eigenvalue $\lambda_h$ approach the true $\lambda$ as the mesh is refined, at the rate the theory predicts? This is a statement about the *discretisation*. - **(B) Is the assembly wired correctly?** — assembly, Dirichlet BCs, and the parametric coefficient with no bug. This is a statement about the *implementation*. The two eyeball figures below answer **(B)** — they are a *wiring smoke test*, not a convergence proof. The [convergence section](#does-the-method-converge) answers **(A)** with a falsifiable rate study, and the [motivating-example overlay](#a-motivating-example) ties both to a first correctness check against a closed-form spectrum. Mesh-independence of the *reduced* model is a separate, later claim. ::: callout-note ## What these curves do and don't show Two axes are in play and the page keeps them apart: - The **space axis is discrete.** $\lambda_h$ is an *approximation* — it sits *near*, not *on*, the analytic value, and the gap only closes as $h \to 0$ (quantified in the [convergence section](#does-the-method-converge)). A single-mesh figure landing close to $(n\pi)^2$ shows the plumbing is sane; it does **not** show the method converges. - The **parameter axis is genuinely continuous.** Each $\lambda_n(\mu)$ is a real, smooth curve of the parametric operator — solved at sample points here, but a continuous object. Smoothness, stable ordering, and the absence of spurious kinks *are* meaningful properties of the parametric problem, and they are drawn correctly. ::: ## 1D parametric Laplacian (test fixture) > Test fixture — not a physics target. Fast 1D scalar problem used to unit-test > the parameter-dimension-agnostic infrastructure cheaply. Do not draw physics > conclusions from it. ### Model On $\Omega = (0, 1)$ with homogeneous Dirichlet boundary conditions, $$ -\bigl(a(\mu)\,u'\bigr)' \;=\; \lambda\,u, \qquad u(0)=u(1)=0, $$ with a piecewise-constant parametric coefficient $a(\mu)$. ### Correctness signature (wiring smoke test) - At $\mu = 0$, the coefficient is $a \equiv 1$ and the leading eigenvalues sit *close to* the analytic spectrum $\lambda_n = (n\pi)^2$ (dashed reference lines) — close, because the space axis is discrete (see the note above). - Each $\lambda_n(\mu)$ is a smooth curve — no kinks or jumps. - Eigenvalue ordering is stable along the parameter path. ![Eigenvalue-vs-µ for the parametric 1D Laplacian. Dashed lines: analytic (nπ)² at µ=0. The curves land *near* the references — a sane-plumbing check, not a convergence proof.](../figures/spectrum_dparam1.png) ## 1D Laplacian, two-parameter coefficient The same fixture exercised with a **two-parameter** coefficient: each panel sweeps one parameter axis with the other held at $0$. This is the *same* 1D Laplacian problem — it is **not** the 2D diffusion model — and its purpose is to smoke-test that the parameter-dimension-agnostic infrastructure (snapshot store, sweep driver, grid) behaves identically as the parameter dimension grows. It precedes the true one-parameter benchmark on the [diffusion-3-3-2 page](diffusion-3-3-2.qmd). ![Eigenvalue-vs-µ for the 1D Laplacian with a two-parameter coefficient (one axis per panel, other axis = 0).](../figures/spectrum_dparam2.png) ## Does the method converge? {#does-the-method-converge} The figures above are single-mesh snapshots; on their own they say nothing about claim **(A)**. The convergence study does. For the 1D Laplacian at $\mu=0$ the exact leading eigenvalue is $\lambda_1 = \pi^2$, so we can refine the mesh ($n_{\text{el}} \in \{25, 50, 100, 200, 400\}$) and watch the discretisation error $|\lambda_{1,h} - \pi^2|$ shrink. ![Log–log mesh convergence of the leading eigenvalue. The data lie on a clean slope-2 line, and every point is *above* the exact value.](../figures/spectrum_convergence.png) Two theory-backed properties hold, and either failing would be a real bug: - **Rate.** The error decays at $O(h^2)$ — the fitted log–log slope is $\approx 2$, exactly the Babuška–Osborn rate for $P_1$ eigenvalue approximation ($O(h^{2k})$ for $P_k$). This is a *falsifiable* statement the single-mesh $(n\pi)^2$ eyeball cannot make. - **From above.** Every $\lambda_{1,h} \ge \lambda_1$. Conforming Galerkin on a subspace $V_h \subset V$ can only *overestimate* eigenvalues (min–max over a smaller space), so monotone convergence from above is the signature of a correctly conforming discretisation. A point dipping *below* $\pi^2$ would signal a non-conforming bug. The rate and the from-above property are both pinned by regression so that a regression in the assembly path is caught rather than silently degrading this page. ## A motivating example {#a-motivating-example} A natural first correctness check is an **exact-vs-FEM overlay**: a model whose spectrum is known in closed form, plotted against the computed FEM spectrum across the whole parameter range. The model is $-\operatorname{div}(A(\mu)\nabla u) = \lambda u$ with $A(\mu) = \operatorname{diag}(1, 1+\mu)$ on the domain $[0,1]^2$ with Dirichlet BCs. The eigenfunctions are $\sin(m\pi x)\sin(n\pi y)$ and the spectrum is closed-form: $$ \lambda_{n,m}(\mu) \;=\; \pi^2\bigl(m^2 + (1+\mu)\,n^2\bigr). $$ ![Exact (lines) vs FEM (dots) for the motivating example, A(µ)=diag(1,1+µ), across the full µ-range. FEM tracks the closed form from above; the curve crossings are what later motivate eigenvalue tracking.](../figures/spectrum_ch2_overlay.png) The FEM dots track the closed-form lines across the entire $\mu$-range, sitting just above them (claim **(A)** again, now on the real 2D physics target rather than the 1D fixture). Two things to notice: - The **eigenfunctions are $\mu$-independent** here (a deliberate simplification of the motivating example), so the only $\mu$-dependence is in the eigenvalues — yet the curves still **cross**. Those crossings are exactly what motivates eigenvalue *tracking* and the reduced-basis rank argument, and they feed directly into the parametric sweep problems built on this substrate. - This overlay is the honest version of "trust the substrate": a model where we know the answer, checked against the code across the whole parameter domain — not a single-mesh glance.