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 averaged rather than pointwise 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 a mild time step restriction $\tau \leqslant ch$ and that the damping parameter in the LLG equations be not too small; these conditions are not needed for the A-stable methods of orders 1 and 2, for which furthermore a discrete energy bound irrespective of solution regularity is obtained.

中文翻译：

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Landau-Lifshitz-Gilbert 方程的高阶线性隐式完全离散化

对于微磁学的 Landau--Lifshitz--Gilbert (LLG) 方程，我们研究了高达 5 阶的线性隐式后向差分公式 (BDF) 时间离散化与高阶非一致有限元空间离散化，这些离散化基于弱由于 Alouges 的公式，但使用由平均而不是逐点正交约束定义的近似切线空间。我们证明了在足够正则解的情况下的稳定性和最优阶误差界限。对于 3 阶到 5 阶的 BDF 方法，这需要温和的时间步长限制 $\tau \leqslant ch$，并且 LLG 方程中的阻尼参数不能太小；对于 1 阶和 2 阶的 A 稳定方法，这些条件不是必需的，此外，获得了与解规则无关的离散能量界。