PH-CON-004

Name: Per-element conservation hotspot

Severity: warning

Input modes: adapter+dump

PH-CON-004 is a 2D mesh-based residual indicator for conservation-defect localization. The rule operates on MeshField inputs from physics_lint[mesh] adapters (scikit-fem 12.0.1+). For each interior element \(K\) of a triangulation, it computes \(\int_K (\Delta_{L^2\text{-proj zero-trace}} u)^2\, dx\) using a P1 finite-element basis and the zero-trace \(L^2\)-projection of the Laplacian. The raw value is the ratio \(\max_K / \mathrm{mean}_K\) — a unitless localization indicator.

The rule fires (APPROXIMATE or FAIL) when the conservation-defect concentrates on a small number of elements relative to the mesh average; typical fired ratios exceed 10. It passes on smooth fixtures where the ratio stays in the 2–5 range, reflecting a well-distributed residual.

Scope — localization, not error bounds. PH-CON-004 is not a Verfürth residual estimator. It omits the \(\|hf\|^2\) source term and the \(\|h^{1/2}[\nabla u \cdot n_e]\|^2\) facet-jump terms, and it does not h-weight per element. Consequently the rule provides neither a reliability estimate (upper bound on \(\|u - u_h\|_{H^1}\)) nor an efficiency estimate (lower bound). It does provide conservation-defect localization: on fixtures where the zero-trace projection is clean, hotspots co-locate with regions of large second-derivative concentration.

The rule emits SKIPPED on non-MeshField inputs (no mesh means no per-element integration) and on PDEs outside its scope. Requires physics-lint[mesh].