# PH-NUM-002

**Name:** Refinement convergence rate below expected

**Severity:** warning

**Input modes:** adapter+dump

PH-NUM-002 detects observed convergence-order on explicitly declared
method-of-manufactured-solutions (MMS) cases for the Laplace equation.
The rule extracts the residual at increasing grid refinements,
computes $\log_2(e_h / e_{h/2})$ using the classical Roy 2005 formula,
and reports `refinement_rate` as the asymptotic observed order. The
expected rate is **case-specific** and depends on PDE, backend, and
boundary treatment — it must be declared up front per case.

Three regimes the rule recognizes:

- **Boundary-dominated FD4 (non-periodic).** The $4N$-cell 2nd-order
  boundary band dominates the $N^2$-cell 4th-order interior at
  $O(h^{2.5})$ on 2D problems. Rule expects rate ≈ 2.5; PASSes within
  $\pm 0.25$.
- **Saturation floor.** Harmonic fixtures on periodic grids (Liouville
  forces constants) or harmonic polynomials on non-periodic FD grids
  (2nd-order FD exact). Both produce residuals below the rule's
  `_SATURATION_FLOOR = 1e-11` and report `rate = inf PASS`.
- **All other cases.** `SKIPPED` (Poisson and heat are not in v1.0
  scope per `ph_num_002.py:92`).

The rule is Laplace-only by v1.0 scope and uses pure numpy — no mesh
assembly, no torch, no scikit-fem. It does not certify convergence
for arbitrary PDE/backend/BC triples; the expected rate must be
declared up front per case.