## Abstract

We show for adaptive triangulations in two dimensions, which are generated by the newest vertex bisection, an optimal grading estimate. Roughly speaking, we construct from the piecewise constant mesh-size function a regularized one with the following two properties. First, the two functions are equivalent, and second, the regularized mesh-size function differs at most by a factor of 2 on neighbouring elements. In combination with Bank & Yserentant (2014, Numer. Math.126, 361–381), this optimal grading estimate enables us to show that the $$L_2$$-orthogonal projections onto the space of continuous Lagrange finite elements up to order 12 is $$H^1$$-stable. We extend these results to a modified red–green refinement.