Axioms of adaptivity for separate marking

Abstract

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 Dörfler marking if ${\mu }^2(\mathcal{T})\le \kappa {\eta }^2(\mathcal{T})$ 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 $\eta (\mathcal{T},K)$ and data approximation errors $\mu (K)$ 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 $H(div)$.

19 pages, 1 figure