Dörfler marking with minimal cardinality is a linear complexity problem

Abstract

Most adaptive finite element strategies employ the Dörfler marking strategy to single out certain elements $\mathcal{M}\subseteq \mathcal{T}$ of a triangulation $\mathcal{T}$ for refinement. In the literature, different algorithms have been proposed to construct $\mathcal{M}$, where usually two goals compete: On the one hand, $\mathcal{M}$ should contain a minimal number of elements. On the other hand, one aims for linear costs with respect to the cardinality of $\mathcal{T}$. Unlike expected in the literature, we formulate and analyze an algorithm, which constructs a minimal set $\mathcal{M}$ at linear costs. Throughout, pseudocodes are given.

20 pages, 1 figure

References in corpus (5)

• Axioms of Adaptivity
• Adaptive FEM with optimal convergence rates for a certain class of non-symmetric and possibly non-linear problems
• Dörfler marking with minimal cardinality is a linear complexity problem
• Rate optimal adaptive FEM with inexact solver for nonlinear operators
• Adaptive BEM with inexact PCG solver yields almost optimal computational costs

Cited by in corpus (16)

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• Plain convergence of adaptive algorithms without exploiting reliability and efficiency
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• The saturation assumption yields optimal convergence of two-level adaptive BEM
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• Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs
• Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
• Optimal multilevel adaptive FEM for the Argyris element