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-N_0 ≤ C M for some absolute constant C C , where N 0 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 n -simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.