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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation
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by Georgios Akrivis, Michael Feischl, Balázs Kovács and Christian Lubich HTML | PDF
Math. Comp. 90 (2021), 995-1038 Request permission


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:
  • 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:
  • Balázs Kovács
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, D-72076 Tübingen, Germany
  • ORCID: 0000-0001-9872-3474
  • Email:
  • Christian Lubich
  • Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, D-72076 Tübingen, Germany
  • MR Author ID: 116445
  • Email:
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 995-1038
  • MSC (2020): Primary 65M12, 65M15; Secondary 65L06
  • DOI:
  • MathSciNet review: 4232216