Axioms of Adaptivity with Separate Marking for Data Resolution

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 $\sigma^2(\mathcal{T},K)=\eta^2(\mathcal{T},K)+\mu^2(K)$ of a finite element domain $K$ in an admissible triangulation $\mathcal{T}$ consists of some residual-based error estimator $\eta(\mathcal{T},K)$ with some reduction property under local mesh-refining and some data approximation error $\mu(K)$. Separate marking means either Dorfler marking if $\mu^2(\mathcal{T})\leq\kappa\eta^2(\mathcal{T})$ or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sufficient conditions on the estimators $\eta(\mathcal{T},K)$ and data approximation errors $\mu(K)$ for optimal asymptotic convergence rates. The enfolded set of axioms of this paper simplifies [C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Comput. Math. Appl., 67 (2014), pp. 1195--1253] for collective m...

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