Mesh independence

Does the adaptive sweep depend on the spatial mesh, or only on the underlying continuous eigenproblem? This page tests that question on the 2-D diffusion eigenproblem with a one-parameter coefficient: the same parameter sweep is run at three mesh resolutions, and the adaptive grid, the reduced-order-model accuracy, and the eigenvector subspaces are compared across them. If the algorithm is discretisation-free in the intended regime, all three resolutions should agree — and the ROM-vs-dense equivalence reported elsewhere should not be an artefact of one particular mesh.

What’s being tested

The one-parameter diffusion sweep is run at three P1 mesh resolutions, \texttt{mesh\_n} \in \{15, 30, 60\}, holding the tolerance tuple fixed: relative pruning threshold t_\pi = 0.21, eigenvalue-cluster threshold t_\lambda = 10^{-3}, and n_{\text{eigs}} = 4 bands tracked per parameter point. Three quantities are compared across the three meshes:

Verdict-tuple convergence — the per-level count tuple (n_{\text{points}}, n_{\text{subintervals}}, n_{\text{uncertified}}) stays stable under refinement, i.e. the adaptive grid lands in the same place at every resolution.
Reduced-order-model accuracy — the relative eigenvalue error of the surrogate against a dense solve stays under a fixed ceiling, the same ceiling reported on the ROM-vs-dense page.
Subspace-angle agreement — the maximum principal angle between matched eigenvector subspaces stays under a fixed ceiling.

Parameters

Quantity	Value
Mesh resolutions	\texttt{mesh\_n} \in \{15,\,30,\,60\}
Parameter box	\mu \in [0.4,\, 1.0]
Pruning / cluster thresholds	t_\pi = 0.21, t_\lambda = 10^{-3}
Eigenvalues per point	n_{\text{eigs}} = 4

Spectra across resolutions

Dense eigenvalue-vs-\mu spectra at the three mesh resolutions (15, 30, 60). The eigenvalue crossing near \mu \approx 0.7 sharpens as the resolution grows, but the converged adaptive grid and the reduced-order-model predictions track it at every resolution.

The converged adaptive grid, the surrogate accuracy, and the eigenvector subspaces all agree across the three resolutions: the algorithm’s behaviour is a property of the continuous eigenproblem, not of the discretisation chosen to solve it.

---
title: "Mesh independence"
---

Does the adaptive sweep depend on the spatial mesh, or only on the underlying
continuous eigenproblem? This page tests that question on the
[2-D diffusion eigenproblem](diffusion-3-3-2.qmd) with a one-parameter
coefficient: the same parameter sweep is run at three mesh resolutions, and the
adaptive grid, the reduced-order-model accuracy, and the eigenvector subspaces
are compared across them. If the algorithm is discretisation-free in the
intended regime, all three resolutions should agree — and the
[ROM-vs-dense equivalence](rom-vs-dense.qmd) reported elsewhere should not be an
artefact of one particular mesh.

## What's being tested

The one-parameter diffusion sweep is run at three P1 mesh resolutions,
$\texttt{mesh\_n} \in \{15, 30, 60\}$, holding the tolerance tuple fixed:
relative pruning threshold $t_\pi = 0.21$, eigenvalue-cluster threshold
$t_\lambda = 10^{-3}$, and $n_{\text{eigs}} = 4$ bands tracked per parameter
point. Three quantities are compared across the three meshes:

1. **Verdict-tuple convergence** — the per-level count tuple
   $(n_{\text{points}}, n_{\text{subintervals}}, n_{\text{uncertified}})$
   stays stable under refinement, i.e. the adaptive grid lands in the same
   place at every resolution.
2. **Reduced-order-model accuracy** — the relative eigenvalue error of the
   surrogate against a dense solve stays under a fixed ceiling, the same
   ceiling reported on the [ROM-vs-dense page](rom-vs-dense.qmd).
3. **Subspace-angle agreement** — the maximum principal angle between matched
   eigenvector subspaces stays under a fixed ceiling.

## Parameters

| Quantity | Value |
|---|---|
| Mesh resolutions | $\texttt{mesh\_n} \in \{15,\,30,\,60\}$ |
| Parameter box | $\mu \in [0.4,\, 1.0]$ |
| Pruning / cluster thresholds | $t_\pi = 0.21$, $t_\lambda = 10^{-3}$ |
| Eigenvalues per point | $n_{\text{eigs}} = 4$ |

## Spectra across resolutions

![Dense eigenvalue-vs-$\mu$ spectra at the three mesh resolutions (15, 30, 60). The eigenvalue crossing near $\mu \approx 0.7$ sharpens as the resolution grows, but the converged adaptive grid and the reduced-order-model predictions track it at every resolution.](../figures/mesh-independence.png)

The converged adaptive grid, the surrogate accuracy, and the eigenvector
subspaces all agree across the three resolutions: the algorithm's behaviour is a
property of the continuous eigenproblem, not of the discretisation chosen to
solve it.