Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1329–1368 | Cite as

Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics

  • Gino Hrkac
  • Carl-Martin Pfeiler
  • Dirk Praetorius
  • Michele RuggeriEmail author
  • Antonio Segatti
  • Bernhard Stiftner
Open Access


We consider the numerical approximation of the Landau–Lifshitz–Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii–Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.


Dzyaloshinskii–Moriya interaction Finite element method Landau–Lifshitz–Gilbert equation Magnetic skyrmions Micromagnetics 

Mathematics Subject Classification (2010)

35K55 65M12 65M60 65Z05 



Open access funding provided by Austrian Science Fund (FWF). The authors thank S. Komineas (University of Crete, Heraklion, Greece) for an informal and stimulating discussion on the topic of this work.

Funding information

This research has been supported by the Vienna Science and Technology Fund (WWTF) through the project Thermally controlled magnetization dynamics (grant MA14-44), by the Austrian Science Fund (FWF) through the doctoral school Dissipation and dispersion in nonlinear PDEs (grant W1245) and the special research program Taming complexity in partial differential systems (grant SFB F65), and by the Engineering and Physical Sciences Research Council (EPSRC) through the projects Picosecond dynamics of magnetic exchange springs (grant EP/P02047X/1) and Coherent spin waves for emerging nanoscale magnonic logic architectures (grant EP/L019876/1).


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© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK
  2. 2.Institute for Analysis and Scientific ComputingTU WienViennaAustria
  3. 3.Faculty of MathematicsUniversity of ViennaViennaAustria
  4. 4.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly

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