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Dörfler marking with minimal cardinality is a linear complexity problem


Authors: Carl-Martin Pfeiler and Dirk Praetorius
Journal: Math. Comp. 89 (2020), 2735-2752
MSC (2010): Primary 65N50, 65N30, 68Q25
DOI: https://doi.org/10.1090/mcom/3553
Published electronically: June 24, 2020
MathSciNet review: 4136545
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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.


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Additional Information

Carl-Martin Pfeiler
Affiliation: TU Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10/E101/4, 1040 Vienna, Austria
MR Author ID: 1323723
Email: carl-martin.pfeiler@asc.tuwien.ac.at

Dirk Praetorius
Affiliation: TU Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10/E101/4, 1040 Vienna, Austria
MR Author ID: 702616
ORCID: 0000-0002-1977-9830
Email: dirk.praetorius@asc.tuwien.ac.at

Keywords: Dörfler marking criterion, adaptive finite element method, optimal complexity
Received by editor(s): July 31, 2019
Received by editor(s) in revised form: February 25, 2020
Published electronically: June 24, 2020
Additional Notes: The authors thankfully acknowledge support by the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245) and through the research project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).
The first author is the corresponding author.
Article copyright: © Copyright 2020 American Mathematical Society