Rate optimality of adaptive finite element methods with respect to overall computational costs
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- by Gregor Gantner, Alexander Haberl, Dirk Praetorius and Stefan Schimanko HTML | PDF
- Math. Comp. 90 (2021), 2011-2040 Request permission
Abstract:
We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.References
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Additional Information
- Gregor Gantner
- Affiliation: Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands
- MR Author ID: 1107233
- ORCID: 0000-0002-0324-5674
- Email: g.gantner@uva.nl
- Alexander Haberl
- Affiliation: TU Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10/E101/4, 1040 Wien, Austria
- MR Author ID: 1147415
- Email: alexander.haberl@asc.tuwien.ac.at
- Dirk Praetorius
- Affiliation: TU Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10/E101/4, 1040 Wien, Austria
- MR Author ID: 702616
- ORCID: 0000-0002-1977-9830
- Email: dirk.praetorius@asc.tuwien.ac.at
- Stefan Schimanko
- Affiliation: TU Wien, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8-10/E101/4, 1040 Wien, Austria
- MR Author ID: 1311878
- Email: stefan.schimanko@asc.tuwien.ac.at
- Received by editor(s): June 6, 2019
- Received by editor(s) in revised form: January 18, 2021
- Published electronically: June 15, 2021
- Additional Notes: The third author is the corresponding author
The authors were supported by the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245), the SFB Taming complexity in partial differential systems (grant F65), and the stand-alone projects Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005) as well as Optimal isogeometric boundary element methods (grant P29096). - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 2011-2040
- MSC (2020): Primary 65N30, 65N50, 65Y20, 65N22, 41A25
- DOI: https://doi.org/10.1090/mcom/3654
- MathSciNet review: 4280291