No . 10 / 2012 On 2 D newest vertex bisection : Optimality of mesh-closure and H 1-stability of L 2-projection
No . 10 / 2012 On 2 D newest vertex bisection : Optimality of mesh-closure and H 1-stability of L 2-projection
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发表时间：
2012
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通讯作者：
J. Kemetmüller
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作者：
J. Schöberl;M. Aurada;M. Feischl;T. Führer;M. Karkulik;J. M. Melenk;J. M. Melenk;A. Parsania;S. Sauter;H. Rezaijafari;H. Woracek;J. Kemetmüller
最新顶点平分法（NVB）是一种常用的局部网格细化策略，适用于由简单体组成的正则三角形剖分。对于二维情况，我们证明了NVB的网格封闭步骤是准最优的，它保留了三角剖分的正则性，并且相应的l -投影在最低阶Courant有限元（P1-FEM）上总是h稳定的。在整个过程中，没有对初始三角测量施加额外的假设。因此，我们的分析改进了Binev， Dahmen & DeVore （number）的结果。数学。97,2004)，卡斯坦森（编者）。大约20,2004)，和史蒂文森(数学。Comp. 77, 2008)，因为他们的定理的所有假设都被删除了。因此，我们的结果放宽了自适应有限元格式在数学上能保证以准最优速率收敛的要求。献给卡斯滕·卡斯滕森，庆祝他50岁生日。
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 L-projection onto lowest-order Courant finite elements (P1-FEM) is always H-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. Dedicated to Carsten Carstensen on the occasion of his 50th birthday.