关于二维最新顶点二分法：网格闭合的最优性与L2投影的H1稳定性
On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H1-Stability of L2-Projection
AI总结：
本文证明了二维最新顶点二分法（NVB）的网格闭合步骤是准最优的，且相应的L2投影到最低阶Courant有限元（P1-FEM）总是H1稳定的，无需对初始三角剖分施加额外假设。
摘要：
最新顶点二分法（NVB）是一种流行的局部网格细化策略，适用于由单纯形组成的正则三角剖分。对于二维情况，我们证明了NVB的网格闭合步骤（保持三角剖分的正则性）是准最优的，并且相应的L2投影到最低阶Courant有限元（P1-FEM）总是H1稳定的。在整个过程中，没有对初始三角剖分施加额外假设。因此，我们的分析改进了Binev、Dahmen和DeVore（Numer. Math. 97, 2004）、Carstensen（Constr. Approx. 20, 2004）以及Stevenson（Math. Comp. 77, 2008）的结果，去除了他们定理中的所有假设。因此，我们的结果放宽了自适应有限元方案在数学上保证以准最优速率收敛所需的条件。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
