Convergence of an Implicit Finite Element Method for the Landau–Lifshitz–Gilbert Equation

Abstract

The Landau-Lifshitz-Gilbert equation describes dynamics of ferromagnetism, where strong nonlinearity, nonconvexity are hard to tackle: so far, existing schemes to approximate weak solutions suffer from severe time-step restrictions. In this paper, we propose an implicit fully discrete scheme and verify unconditional convergence.1991 Mathematics Subject Classification. 35K55, 65M12, 65M15, 68U10, 94A08.

Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

“…In [3,4], a fully practical implicit scheme is given to solve problem (A), which is based on a reformulation of (1.2) using cross products (m = 3),…”

Constraint Preserving, Inexact Solution of Implicit Discretizations of Landau–Lifshitz–Gilbert Equations and Consequences for Convergence

“…P r o o f. We sketch the proof of the theorem and refer the reader to [4] for details. Given any T such that Jk > T > 0, summation of the estimate in Lemma 2.1 over j = 0, 1, ..., J shows…”

“…A fixed-point iteration that converges to a solution of the nonlinear system of equations (3) provided that k = O(h 2 ) is given in [4]. However, the analysis in [4] has two drawbacks: (i) the global convergence analysis towards a weak solution of (1) requires the exact solution of (3) and (ii) iterates in the fixed-point iteration do not satisfy the unit-length constraint at the nodes. In this paper we aim at improving these two points.…”

“…These conditions on the numerical parameters become very restrictive if n = 3 or if α is small. An implicit discretization that conserves the unit-length constraint and thereby avoids any projection steps and that is unconditionally stable and convergent has been designed in [4]. It is based on the equivalent Gilbert form of (1),…”

A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

“…Their work is extended in [10] to harmonic map heat flow into non-constant target manifolds. Generalizations to p-harmonic heat flow and the Maxwell-Landau-Lifshitz-Gilbert equations are given in [3,9] and [1,8], respectively. The related problem of wave map into sphere, which has applications to general relativity, is considered in [6,7].…”