On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H1-Stability of L2-Projection

TL;DR

This paper proves that in 2D, the mesh-closure step of newest vertex bisection is quasi-optimal and ensures H1-stability of the L2-projection onto P1 finite elements, without extra initial assumptions.

Contribution

It removes previous assumptions on initial triangulations, improving theoretical guarantees for adaptive finite element methods in 2D.

Findings

Mesh-closure step of NVB is quasi-optimal.

L2-projection onto P1 elements is H1-stable.

Results hold without initial triangulation assumptions.

Abstract

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 mesh-closure 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.