The propagation problem in longest-edge refinement

Two asymptotic properties that arise in iterative mesh refinement of triangles are introduced and investigated. First, we provide theoretical results showing that recursive application of uniform four triangles longest-edge (4T-LE) partition to an arbitrary unstructured triangular mesh produces meshes in which the triangle pairings sharing a common longest edge asymptotically tend to cover the area of the whole mesh. As a consequence, we prove that for a triangle, the induced exterior conforming refinement zone extends on average to a few neighbor adjacent triangles. We determine the asymptotic extent of this propagating path and include results of supporting numerical experiments with uniform and adaptive mesh refinement. Similar behavior and LE propagation from a four triangle self similar (4T-SS) local subdivision alternative is analyzed and compared numerically. Hybrid 4T-LE and 4T-SS LE schemes are also considered. The results are relevant to mesh refinement in finite element and finite volume calculations as well as mesh enhancement in Computer Graphics and CAGD.