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

Abstract

Abstract. The Landau-Lifshitz-Gilbert equation describes magnetic behavior in ferromagnetic materials. Construction of numerical strategies to approximate weak solutions for this equation is made difficult by its top order nonlinearity and nonconvex constraint. In this paper, we discuss necessary scaling of numerical parameters and provide a refined convergence result for the scheme first proposed by Alouges and Jaisson (2006). As an application, we numerically study discrete finite time blowup in two dimensions.

A finite-difference scheme for a model of magnetization dynamics with inertial effects

“…We limit ourselves to mentioning a handful of references and refer the interested reader to the survey [6] for a more detailed bibliographical account. The general framework (although without inertial effects, i.e., the case ζ = 0) was established in earlier papers by finite-difference/element methods; see, for instance, [7][8][9][10][11]. The following results concern systems coupling the LLG equation with the Maxwell system [4,[12][13][14][15][16].…”

On a Nonlocal Damping Model in Ferromagnetism

“…Several schemes were proposed and their convergence to weak solutions was proved. A significant step forward in the convergence theory of numerical schemes has been done recently; see [19][20][21]. This will be helpful to give a strategy for efficient computer implementation which may reflect the true nature of the augmentation of the LLG model considered in this paper.…”

A decoupled and unconditionally convergent linear FEM integrator for the Landau-Lifshitz-Gilbert equation with magnetostriction

“…This choice of m 0 is motivated in Bartels et al (2008), in order to form some singularity at the centre x = 0. The triangulation T r used in the numerical simulation is defined through a positive integer r and consists of 2 2r+1 halved squares with edge length h = 2 −r .…”

“…For a review of the analysis of LLG, we refer to Kružík & Prohl (2006), García-Cervera (2007) and Cimrak (2008) or the monographs (Hubert & Schäfer, 1998;Prohl, 2001) and the references therein. As far as the numerical analysis is concerned, mathematically reliable and convergent LLG integrators are found in Bartels & Prohl (2006), Alouges (2008), Baňas et al (2008), Bartels et al (2008), Alouges et al (2011), Goldenits et al (2011), Bruckner et al (2012), Goldenits et al (2012), Le & Tran (2012), Rochat (2012) and Baňas et al (2013). Of particular interest are unconditionally convergent integrators, which do not impose a coupling of spatial mesh size h and timestep size k to ensure stability of the numerical scheme.…”