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

Akrivis, Georgios; Feischl, Michael; Kovács, Balázs; Lubich, Christian

doi:10.1090/mcom/3597

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;

Math. Comp. 90 (2021), 995-1038

DOI: https://doi.org/10.1090/mcom/3597

Published electronically: February 26, 2021

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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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