Justification of asymptotic stability of the triangulation algorithm for a three-dimensional domain

An algorithm of triangulation construction (the subdivision into tetrahedrons) for a three-dimensional bounded domain with a smooth curvilinear boundary is considered. The algorithm starts on a given coarsest triangulation. The consequent finer triangulations are recurrently constructed by the subdivision of tetrahedrons of the previous level into 8 parts with correction of the location of vertices near the boundary to approximate the boundary. To evaluate the quality of a triangulation a certain quantitative criterion is used. It is proved that a successful (in the sense of this criterion) initial triangulation moderately detailed guarantees good quality of the consequent finer triangulations under arbitrary number of recurrent implementations of subdivision algorithm.