Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism

Numerical analysis of an explicit approximation scheme for the Landau-Lifshitz-Gilbert equation

“…In this version of the LLG equation, ∆u is a simple approximation of an effictive field which is in more general models replaced by H eff (u) = −∇ L 2 E LL (u), where E LL is the Landau-Lifshitz energy of micromagnetics; cf., e.g., [11]. The Cauchy problem for LLG with natural boundary conditions, then, is the problem of finding u, given initial data u 0 : Ω ⊆ R n → S 2 satisfying We will refer to the first term as the gyroscopic term and the second as the damping term.…”

“…Recent remedies have been made, partially addressing the dual requirements of efficiency and reliability: (i) projection methods have been constructed [6,7,16], independently dealing with the nonconvex algebraic constraint; however, no (discrete) energy principle is available, and convergence to LLG is only known in the case of existing strong solutions to LLG; (ii) explicit/implicit discretizations of Ginzburg-Landau penalizations that involve an additional parameter ε > 0 are used, which allow for a discrete energy principle, possibly for restricted choices of spatio-temporal discretization parameters. We refer to [11] for a more detailed discussion in this direction. Alouges and Jaisson [1] propose a finite element plus projection scheme, which is shown to converge if successively the time-step size and the mesh size tend to zero.…”

On certain convex compactifications for relaxation in evolution problems

“…e.g. [2,6,18,22,23,24,29,32,39]. This is a relatively standardly used approach, especially if a correlation of microstructures between particular time instances is rather "macroscopical" only.…”

Stabilization methods in relaxed micromagnetism

“…Alternatively, we deal with the following stabilization ansatzes, 13) with β E | E = β E > 0. Finally, other numerical scheme fit into this framework that employ element-wise affine, globally continuous magnetizations,…”

“…Hysteretic behavior can be modeled by Landau-Lifshitz-Gilbert equation [12,17], which is a mesoscopic (phenomenological) model to describe spin dynamics behavior on submicron spatial scales across no more than some nanoseconds [13]; in contrast, macroscopic (phenomenological) models describe evolutionary behavior in ferromagnetic bulk specimen up to millimeter size on time-scales of much larger magnitude. Recently, a new evolutionary, rate-independent macroscopic model to describe hysteresis losses based on plasticity models in metals and shape-memory alloys has been proposed by Kružík and Roubíček [14]: here, dissipation/hysteretic effects are described in terms of evolution of the configuration, t → q(t) ≡ ( ν ν ν(t), m(t) ), where at positive times configurations…”