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

For the Landauâ??Lifshitzâ??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 weak formulation due Alouges but use approximate tangent spaces that defined by $L^2$-averaged instead nodal orthogonality constraints. We prove stability and optimal-order error bounds in situation a sufficiently regular solution. BDF methods orders $3$ $5$, this requires damping parameter LLG equations be above positive threshold; condition is not needed for A-stable $1$ $2$, furthermore discrete energy inequality irrespective solution regularity proved.