Mesh Refinement Processes Based on the Generalized Bisection of Simplices

Mesh-refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of Rheinboldt [SIAM J. Numer. Anal., 17 (1980), pp. 766–778]. Then it is proved that such processes allow the construction of sequences of naturally smooth, conforming and nested nonuniform meshes. In fact it is shown that there exists a generalized mesh-refinement operator that for any conforming triangulation Δ and for any V ⊂ Δ, produces a nested, smooth, conforming triangulation Δ ^ * containing successors of all elements of V and such that the minimum cell-size of Δ ^ * is bounded from below by half the minimum cell-size of Δ. Two conforming mesh-refinement algorithms that allow the selective refinement of computational triangulations are explicitly given.