Local bisection refinement for $n$-simplicial grids generated by reflection

Abstract

A simple local bisection refinement algorithm for the adaptive refinement of $n$-simplicial grids is presented. The algorithm requires that the vertices of each simplex be ordered in a special way relative to those in neighboring simplices. It is proven that certain regular simplicial grids on $[0,1]^n$ have this property, and the more general grids to which this method is applicable are discussed. The edges to be bisected are determined by an ordering of the simplex vertices, without local or global computation or communication. Further, the number of congruency classes in a locally refined grid turns out to be bounded above by $n$, independent of the level of refinement. Simplicial grids of higher dimension are frequently used to approximate solution manifolds of parametrized equations, for instance, as in [W. C. Rheinboldt, Numer. Math., 53 (1988), pp. 165–180] and [E. Allgower and K. Georg, Utilitas Math., 16 (1979), pp. 123–129]. They are also used for the determination of fixed points of functions from ${\bf R}^n$ to ${\bf R}^n$, as described in [M. J. Todd, Lecture Notes in Economic and Mathematical Systems, 124, Springer-Verlag, Berlin, 1976]. In two and three dimensions, such grids of triangles, respectively, tetrahedrons, are used for the computation of finite element solutions of partial differential equations, for example, as in [O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, 1984], [R. E. Bank and B. D. Welfert, SIAM J. Numer. Anal., 28 (1991), pp. 591–623], [W. F. Mitchell, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 146–147], and [M. C. Rivara, J. Comput. Appl. Math., 36 (1991), pp. 79–89]. The new method is applicable to any triangular grid and may possibly be applied to many tetrahedral grids using additional closure refinement to avoid incompatibilities