Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018) proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schr\"odinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties.
翻译:最近,Kim & Wilkening (Landau-Lifshitz 等式的大规模拉动有限元素法的说服力,Quart. Appl. Math., 76, 383-405, 2018) 提议了两种新型的微磁学Landau-Lifshitz-Gilbert 等式预测器(LLLG) 的预测器校正器方法,该等式模拟了铁磁材料磁化的动态。 这两种整合器都以所谓的LLADau- Lifshitz 等式LLG为基础, 使用按一级定式分解的大规模拉动变异配方, 尽管LLLG非线性化, 76, 383-405, 2018), 但只要求线性系统的解决办法。 第一个(时间顺序) 方法将线性更新Landau- Lifshit- Lifshit 等式的预测器(LLLLG- Inher) 等式。 在第二( 级( 级( 级) 等式中, 等式中, 投法中, 投影化步骤被一个线限制的线性变换换换成一个线- 。