Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation

Abstract

The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.

Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics

“…Again, these claims are confirmed in our numerical studies. Stability and convergence of PC2, not addressed in [KW18], remain open also in our analysis and will be the subject of future research. In this paper, we shed some light on this question by means of some surprising numerical experiments.…”

“…In this section, we discuss the first-order scheme proposed in [KW18] and its connections with the integrators proposed in [BP06] and [Alo08]. Our contribution is twofold: First, we prove unconditional well-posedness of the scheme, which fills a fundamental gap in the analysis of [KW18].…”

“…In this work, we improve the theoretical understanding of the predictor-corrector methods proposed in [KW18].…”

“…In this section, we discuss the first-order scheme proposed in [KW18] and its connections with the integrators proposed in [BP06] and [Alo08]. Our contribution is twofold: First, we prove unconditional well-posedness of the scheme, which fills a fundamental gap in the analysis of [KW18]. Second, we employ an explicit treatment of the (nonlocal) lower-order contributions to obtain a computationally superior IMEX version of the scheme, preserving (unconditional) convergence and experimental rates in time.…”

“…The recent work [KW18] proposes two predictor-corrector schemes for LLG which aim to combine the features of some of the above integrators. In the first scheme, [KW18, Algorithm 1], which we denote by PC1 for the sake of brevity, the predictor is based on the LL form (1) of LLG (like the variational formulation used in [Cim09]) and employs mass-lumping for its discretization (like [BP06,Cim09]).…”