Grading of Triangulations Generated by Bisection

For triangulations generated by the adaptive bisection algorithm by Maubach and Traxler we prove existence of a regularized mesh function with grading two. This sharpens previous results in two dimensions for the newest vertex bisection and generalizes them to arbitrary dimensions. In combination with Diening et al. (2021) this yields $H^1$-stability of the $L^2$-projection onto Lagrange finite element spaces for all polynomial degrees and dimensions smaller than seven.