HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION OF THE LANDAU-LIFSHITZ-GILBERT EQUATION

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.

Automated summary

This study focuses on the time discretization of the Landau-Lifshitz-Gilbert (LLG) equation in micromagnetics, proving stability and error bounds in the situation of a sufficiently regular solution. A positive damping parameter threshold is required for higher-order BDF methods, while A-stable methods do not have this requirement and also demonstrate a discrete energy inequality regardless of solution regularity.