The completion of locally refined simplicial partitions created by bisection

Recently, in [Found. Comput. Math., 7(2) (2007), 245-269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466-488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles $N$ in the output partition of such a method is generally larger than the number $M$ of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes.A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219-268] by Binev, Dahmen and DeVore saying that $N-N_0 \leq C M$ for some absolute constant $C$, where $N_0$ is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of $n$-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.