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)(T, K)=eta(2)(T, K)+mu(2)(K) of a finite element domain K in an admissible triangulation T consists of some residual-based error estimator eta(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)(T) <= kappa eta(2)(T) or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sufficient conditions on the estimators eta(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 marking, treats separate marking established for the first time in an abstract framework, generalizes [C. Carstensen and E.-J. Park, SIAM J. Numer. Anal., 53 (2015), pp. 43-62] for least-squares schemes, and extends [C. Carstensen and H. Rabus, Math. Comp., 80 (2011), pp. 649-667] to the mixed finite element method with flux error control in H(div). The paper gives an outline of the mathematical analysis for optimal convergence rates but also serves as a reference so that future contributions merely verify a few axioms in a new application in order to ensure optimal mesh-refinement of the adaptive algorithm.