Akrivis, G., Feischl, M., Kovács, B., & Lubich, C. (2021). HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION OF THE LANDAU–LIFSHITZ–GILBERT EQUATION. Mathematics of Computation, 90(329), 995–1038. https://doi.org/10.1090/mcom/3597
BDF methods; energy technique; Landau— Lifshitz-Gilbert equation; non-conforming finite element method; stability
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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 L2-averaged instead of nodal orthogonality constraints. We prove stability and optimalorder 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.
