BDF2-Type Integrator for Landau-Lifshitz-Gilbert Equation in Micromagnetics: A-Priori Error Estimates

Jul 16, 2026

This paper studies the Landau–Lifshitz–Gilbert (LLG) equation modeling time-dependent micromagnetic dynamics. It analyzes a fully discrete method that couples first-order finite elements in space with a BDF2 scheme in time, requiring only one linear system solve per time step and not enforcing the pointwise unit-length constraint on magnetization. Building on prior weak-convergence results, it proves optimal-order convergence rates for sufficient regularity of the exact solution and external field. Numerical tests show first-order spatial and second-order temporal convergence.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages37

Reading Time1 h 33 min

Introduction

Background and modeling of the LLG equation

Numerical discretization approaches: weak vs strong convergence

Prior tangent-plane and projection-free integrators

Goal and contribution of the BDF2 higher-order time integrator

arXiv :2605 .05129v3 [math .NA] 10 Jul 2026 BDF2-TYPE INTEGRATOR FOR LANDAU-LIFSHITZ-GILBERT EQUATION IN MICROMAGNETICS: A-PRIORI ERROR ESTIMATES MICHELE ALDÉ, MICHAEL FEISCHL, AND DIRK PRAETORIUS Abstract. We consider the Landau–Lifshitz–Gilbert equation (LLG), which models timedependent micromagnetic phenomena. We analyze a fully discrete scheme that combines firstorder finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In combination with our previous work, this establishes the first linear higher-order-in-time integrator that converges both to weak and strong solutions of LLG. Numerical experiments confirm first-order convergence in space and second-order convergence in time. 1. Introduction Time-dependent micromagnetic phenomena are typically modeled by the phenomenological Landau–Lifshitz–Gilbert equation (LLG), a nonlinear time-dependent PDE that describes the evolution of the vector field magnetization m (t, x) ∈ R3 in a ferromagnetic body Ω ⊂ Rd , d = 2 , 3. The magnetization represents the magnetic moment per unit volume and can be interpreted as the continuum counterpart of discrete magnetic spins. It is a fundamental physical property of the model that, for constant temperatures below the Curie temperature, the modulus of the magnetization remains constant, i.e. , |m(t, x)| = Ms, where Ms > 0 denotes the saturation magnetization. The equation plays a central role in micromagnetics and underlies many areas such as sensors, actuators, memory storage devices such as hard disk drives and magnetic recording [LL35 ; Bro62 ; Bro63 ; HS98] . The analytical theory of LLG is by very well-developed. We refer to [AS92] for the global-intime existence of weak solutions, to [CDG98 ; CF01 ; Cim05 ; Cim07 ; FT17a; GX23] for local-intime existence of smooth strong solutions, and to [DS14 ; DIP20] for weak-strong uniqueness, i.e. , if a strong solution exists until a certain time, then every weak solution with the same initial data also coincides with it until that time. In particular, strong solutions are unique if they exist. From the point of view of numerical analysis, discretization schemes of LLG may be grouped into schemes that converge, at a given rate, to sufficiently smooth strong solutions under sufficient regularity assumptions on the exact solution [Pro01 ; BS06 ; Gao14 ; An16 ; AFKL21 ; AS22 ; CCWX22 ; CCWX23 ; LHH23] and schemes that only guarantee weak convergence, along asubsequence, to weak solutions of LLG [AJ06; BKP08 ; BP06 ; Alo08 ; Rug22] . Some works also turn attention to coupled LLG systems [BPPR14; BPP15 ; DPP+20 ; AKT12 ; BSF+14 ; PRS18 ; HPP+19 ; DPPR23 ; GJL+26], employing an implicit-explicit (IMEX) discretization to reduce the computational cost. One of the first works that proved unconditional weak convergence to weak solutions of LLGis [Alo08], where a projection-based first-order in time tangent-plane integrator is proposed and analyzed. The main idea of the scheme is to exploit inherited properties of the equation to turn the non-linear PDE in the variable m into a linear variational formulation for the variable Acknowledgment. This research was funded by the Austrian Science Fund (FWF) projects 10.55776/F65 (SFB F65 “Taming complexity in PDE systems”), and 10.55776/P33216 (stand-alone project P33216 “Computational nonlinear PDEs”) and through the ERC grant 10.3030/101125225 “New frontiers in optimal adaptivity” of Michael Feischl. Additionally, the Vienna School of Mathematics supports Michele Aldé . v := ∂tm, so that only one linear system is solved at every time-step of the algorithm. This integrator relies

FAQs

How do the numerical experiments validate the method?

The experiments confirm first-order convergence in space and second-order convergence in time, matching the expected accuracy of the finite element spatial discretization and the BDF2 temporal integrator.

Jul 16, 2026

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