Issue |
ESAIM: M2AN
Volume 59, Number 1, January-February 2025
|
|
---|---|---|
Page(s) | 167 - 199 | |
DOI | https://doi.org/10.1051/m2an/2024065 | |
Published online | 14 January 2025 |
Convergent finite element methods for antiferromagnetic and ferrimagnetic materials
1
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK
2
Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
* Corresponding author: m.ruggeri@unibo.it
Received:
7
December
2023
Accepted:
19
August
2024
We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations averaging the spins of two sublattices. For the static setting, which requires the solution of a constrained energy minimization problem, we introduce a discretization based on first-order finite elements and prove its Γ-convergence. Then, we propose and analyze two iterative algorithms for the computation of low-energy stationary points. The algorithms are obtained from (semi-)implicit time discretizations of gradient flows of the energy. Finally, we extend the algorithms to the dynamic setting, which consists of a nonlinear system of two Landau–Lifshitz–Gilbert equations solved by the two fields, and we prove unconditional stability and convergence of the finite element approximations toward a weak solution of the problem. Numerical experiments assess the performance of the algorithms and demonstrate their applicability for the simulation of physical processes involving antiferromagnetic and ferrimagnetic materials.
Mathematics Subject Classification: 35K61 / 65M12 / 65M60 / 65Z05
Key words: Antiferromagnetism / ferrimagnetism / finite element method / Γ-convergence / Landau–Lifshitz–Gilbert equation
© The authors. Published by EDP Sciences, SMAI 2025
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.