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 N in the output partition of such a method is generally larger than the number M 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 ≤ C M N-N₀ C M for some absolute constant C C, where N 0 N₀ 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 n -simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.