Abstract:The Landau--Lifshitz--Gilbert (LLG) equation is a fundamental model in micromagnetics for the magnetisation dynamics of ferromagnets at temperatures well below the Curie temperature. In chiral magnetic materials, the bulk Dzyaloshinskii--Moriya interaction (DMI) contributes a first-order term to the effective field, induces a natural chiral boundary condition, and plays an important role in the dynamics of skyrmions. In this paper, we analyse a mass-lumped midpoint finite element scheme for the LLG equation with exchange interaction and bulk DMI. The fully discrete scheme preserves the nodal unit-length constraint exactly and satisfies a discrete energy law. Under suitable regularity assumptions on the exact solution, we prove an optimal-order convergence estimate in the energy norm, first order in space and second order in time. To our knowledge, this is the first such a priori error estimate for the mass-lumped midpoint finite element method, even in the exchange-only case. Since the midpoint scheme is nonlinear, its practical implementation requires an algebraic solver that is compatible with the geometric structure of the method. To this end, we propose a structure-preserving fixed-point iteration which preserves the nodal unit-length constraint at every iterate. We further prove that, under an appropriate residual stopping criterion, the resulting inexact scheme retains the same convergence rate as the ideal midpoint scheme, up to the contribution of the solver tolerance. Numerical experiments support the theoretical results and illustrate the robustness and structure-preserving properties of the method.
From: Agus Soenjaya [view email]
[v1]
Mon, 15 Jun 2026 02:58:40 UTC (4,268 KB)
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