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M. Karkulik, D. Pavlicek, D. Praetorius:
"On 2D newest vertex bisection: Optimality of mesh-closure and H^1-stability of L_2-projection";
Constructive Approximation, 38 (2013), 2; S. 213 - 234.

Newest vertex bisection (NVB) is a popular local mesh-refinement
strategy for regular triangulations which consist of simplices. For
the 2D case, we prove that the meshclosure step of NVB, which
preserves regularity of the triangulation, is quasi-optimal and that
the corresponding L2-projection onto lowest-order Courant finite
elements (P1-FEM) is always H1-stable. Throughout, no additional
assumptions on the initial triangulation are imposed. Our analysis
thus improves results of Binev, Dahmen & DeVore (Numer. Math. 97,
2004), Carstensen (Constr. Approx. 20, 2004), and Stevenson (Math.
Comp. 77, 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.

adaptive finite element methods, regular triangulations, newest vertex bisection, L^2-projection, H^1-stability

http://dx.doi.org/10.1007/s00365-013-9192-4