On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection

Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations that consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L (2)-projection onto lowest-order Courant finite elements (P1-FEM) is always H (1)-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev et al. (Numer. Math. 97(2):219-268, 2004), Carstensen (Constr. Approx. 20(4):549-564, 2004), and Stevenson (Math. Comput. 77(261):227-241, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.