Mechanisms — phenomena and algorithmic responses

Two pages that explain why something non-obvious happens and what the algorithm does about it. Neither page is a run on a new problem; both are arguments about the structure of the method.

Pages

Persistent Π failures at eigenvalue crossings

The a-posteriori matrix \Pi is the algorithm’s certification signal. On a well-resolved subinterval it is diagonally dominant; on an ambiguous one it flags REFINE. At a true eigenvalue crossing, \Pi is persistently off-diagonal at every refinement level — no amount of bisection removes the flag. This is a topological property of the eigenvalue surface, not a solver artifact. The page traces the exact mechanism and explains why it is the precise reason the Projection Algorithm is needed and what the Projection Algorithm does.

Dynamic domain decomposition

The global snapshot store forces a single reduced basis that spans every regime the adaptive grid has visited. When the eigensurface is only piecewise low-rank — a smooth background plus concentrated bursts near eigenvalue crossings — the global basis rank grows as the sum of per-regime ranks, not the maximum. Dynamic parameter-domain decomposition fixes this by splitting the parameter box into regions, each with its own snapshot store, so the peak per-region rank approaches “max over regimes” instead of “sum over regimes.”