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Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation


Authors: Georgios Akrivis, Michael Feischl, Balázs Kovács and Christian Lubich
Journal: Math. Comp. 90 (2021), 995-1038
MSC (2020): Primary 65M12, 65M15; Secondary 65L06
DOI: https://doi.org/10.1090/mcom/3597
Published electronically: February 26, 2021
MathSciNet review: 4232216
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Abstract: For the Landau–Lifshitz–Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order $5$ combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by $L^2$-averaged instead of nodal orthogonality constraints. We prove stability and optimal-order error bounds in the situation of a sufficiently regular solution. For the BDF methods of orders $3$ to $5$, this requires that the damping parameter in the LLG equations be above a positive threshold; this condition is not needed for the A-stable methods of orders $1$ and $2$, for which furthermore a discrete energy inequality irrespective of solution regularity is proved.


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Additional Information

Georgios Akrivis
Affiliation: Department of Computer Science & Engineering, University of Ioannina, 451 10 Ioannina, Greece; Institute of Applied and Computational Mathematics, FORTH, 700 13 Heraklion, Crete, Greece
MR Author ID: 24080
Email: akrivis@cse.uoi.gr

Michael Feischl
Affiliation: Institute for Analysis and Scientific Computing (E 101), Technical University Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
MR Author ID: 965785
Email: michael.feischl@tuwien.ac.at

Balázs Kovács
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, D-72076 Tübingen, Germany
ORCID: 0000-0001-9872-3474
Email: kovacs@na.uni-tuebingen.de

Christian Lubich
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, D-72076 Tübingen, Germany
MR Author ID: 116445
Email: lubich@na.uni-tuebingen.de

Keywords: BDF methods, non-conforming finite element method, Landau–Lifshitz–Gilbert equation, energy technique, stability
Received by editor(s): March 13, 2019
Received by editor(s) in revised form: March 14, 2020
Published electronically: February 26, 2021
Additional Notes: The work of the second, third, and fourth authors were supported by Deutsche Forschungsgemeinschaft – Project-ID 258734477 – SFB 1173.
Article copyright: © Copyright 2021 American Mathematical Society