适应性公理：分离标记的公理
Axioms of adaptivity for separate marking
AI总结：
本文提出分离标记的公理，用于混合有限元方法和div-最小二乘有限元方法的自适应网格细化，通过残差误差估计和数据近似误差的组合，确保最优渐近收敛率。
摘要：
具有$H(div)$范数的流误差的混合有限元方法和div-最小二乘有限元方法需要在强制自适应网格细化中采用分离标记策略。有限元域$K$在允许的三角剖分$\mathcal T$中的细化指示符$σ^2(\mathcal T,K)=η^2(\mathcal T,K)+μ^2(K)$由某些基于残差的误差估计器$η(\mathcal T,K)$和某些在局部网格细化下的减少性质以及某些数据近似误差$μ(K)$组成。分离标记意味着如果$μ^2(\mathcal T) \leq κη^2(\mathcal T)$则为Dörfler标记，否则则运行一个最优的数据近似算法，其精度受[Carstensen, Rabus, Math.Comp. 2011; Rabus, J.Numer.Math. 2015]中建立的控制。这些公理是误差估计器$η(\mathcal T,K)$和数据近似误差$μ(K)$的抽象和充分条件，用于实现最优渐近收敛率。包含的公理集简化了[CFP14]中的集体标记，以抽象框架处理首次提出的分离标记，扩展了[Carstensen, Park, SIAM J.Numer.Anal. 2015]对最小二乘方案的应用，并将[Carstensen, Rabus, Math.Comp. 2011]扩展到具有流误差控制的混合FEM中。
英文摘要：
Mixed finite element methods with flux errors in $H(div)$-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator $σ^2(\mathcal T,K)=η^2(\mathcal T,K)+μ^2(K)$ of a finite element domain $K$ in an admissible triangulation $\mathcal T$ consists of some residual-based error estimator $η(\mathcal T,K)$ with some reduction property under local mesh-refining and some data approximation error $μ(K)$. Separate marking means either Dörfler marking if $μ^2(\mathcal T) \leq κη^2(\mathcal T)$ or otherwise an optimal data approximation algorithm runs with controlled accuracy as established in [Carstensen, Rabus, Math.Comp. 2011; Rabus, J.Numer.Math. 2015]. The axioms are abstract and sufficient conditions on the estimators $η(\mathcal T,K)$ and data approximation errors $μ(K)$ for optimal asymptotic convergence rates. The enfolded set of axioms simplifies \cite{CFP14} for collective marking, treats separate marking established for the first time in an abstract framework, generalizes [Carstensen, Park, SIAM J.Numer.Anal. 2015] for least-squares schemes, and extends [Carstensen, Rabus, Math.Comp. 2011] to the mixed FEM with flux error control in $H(div)$.
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