Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau–Lifshitz equation, Quart. Appl. Math., 76, 383–405, 2018) proposed two novel predictor-corrector methods for the Landau–Lifshitz–Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau–Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.

Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics / Mauser N.J.; Pfeiler C.-M.; Praetorius D.; Ruggeri M.. - In: APPLIED NUMERICAL MATHEMATICS. - ISSN 0168-9274. - ELETTRONICO. - 180:(2022), pp. 33-54. [10.1016/j.apnum.2022.05.008]

Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics

Ruggeri M.
2022

Abstract

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau–Lifshitz equation, Quart. Appl. Math., 76, 383–405, 2018) proposed two novel predictor-corrector methods for the Landau–Lifshitz–Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau–Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.
2022
Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics / Mauser N.J.; Pfeiler C.-M.; Praetorius D.; Ruggeri M.. - In: APPLIED NUMERICAL MATHEMATICS. - ISSN 0168-9274. - ELETTRONICO. - 180:(2022), pp. 33-54. [10.1016/j.apnum.2022.05.008]
Mauser N.J.; Pfeiler C.-M.; Praetorius D.; Ruggeri M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/944115
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