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A hybrid boundary integral-PDE approach for the approximation of the demagnetization potential in micromagnetics

Doghonay Arjmand, Victor Martinez Calzada

TL;DR

This work introduces a hybrid boundary-integral–PDE framework to efficiently approximate the micromagnetic demagnetization potential, replacing the expensive volume integral with two uncoupled PDEs solved on a bounded domain and a single-layer boundary integral. The method achieves near-linear scaling by solving a regularised interior PDE and a boundary integral problem separately, enabling parallel computation. Theoretical analysis establishes exponential convergence with respect to domain truncation under periodic or high-frequency assumptions and provides explicit error bounds. Numerical experiments in two dimensions validate the convergence rates and show qualitative agreement with volume-integral baselines, highlighting the approach's practicality for large-scale micromagnetic simulations.

Abstract

The demagnetization field in micromagnetism is given as the gradient of a potential which solves a partial differential equation (PDE) posed in R^d. In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem relies on the representation of the potential via the Green's function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green's function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs is obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings; periodic magnetisation, and high-frequency magnetisation. Numerical examples are given to verify the convergence rates.

A hybrid boundary integral-PDE approach for the approximation of the demagnetization potential in micromagnetics

TL;DR

This work introduces a hybrid boundary-integral–PDE framework to efficiently approximate the micromagnetic demagnetization potential, replacing the expensive volume integral with two uncoupled PDEs solved on a bounded domain and a single-layer boundary integral. The method achieves near-linear scaling by solving a regularised interior PDE and a boundary integral problem separately, enabling parallel computation. Theoretical analysis establishes exponential convergence with respect to domain truncation under periodic or high-frequency assumptions and provides explicit error bounds. Numerical experiments in two dimensions validate the convergence rates and show qualitative agreement with volume-integral baselines, highlighting the approach's practicality for large-scale micromagnetic simulations.

Abstract

The demagnetization field in micromagnetism is given as the gradient of a potential which solves a partial differential equation (PDE) posed in R^d. In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to solve this problem relies on the representation of the potential via the Green's function, where a volume and a boundary integral terms need to be accurately approximated. From a computational point of view, the volume integral dominates the computational cost and can be difficult to approximate due to the singularities of the Green's function. In this article, we propose a hybrid model, where the overall potential can be approximated by solving two uncoupled PDEs posed in bounded domains, whereby the boundary conditions of one of the PDEs is obtained by a low cost boundary integral. Moreover, we provide a convergence analysis of the method under two separate theoretical settings; periodic magnetisation, and high-frequency magnetisation. Numerical examples are given to verify the convergence rates.
Paper Structure (14 sections, 10 theorems, 90 equations, 5 figures)

This paper contains 14 sections, 10 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivation of a known integral formulation
  4. A Hybrid Approach
  5. Analysis
  6. Effect of truncating the infinite domain in \ref{['Eqn_InfDomain_Elliptic']}
  7. Analysis of an infinitely large periodic magnet
  8. Analysis for magnetization with vanishing frequency components in the origin
  9. Numerical Approximations
  10. Approximation of the regularised PDE \ref{['Eqn_RegularisedElliptic']}
  11. Numerical approximation of the boundary integral
  12. Numerical results
  13. A qualitative comparison of the approximation \ref{['Eqn_Approximation']}
  14. Convergence study

Key Result

Lemma 1

Suppose $u$ satisfies equation Eqn_Demag along with the boundary conditions, then $u$ satisfies the integral representation Eqn_Integral_Representation.

Figures (5)

  • Figure 1: Transformation to polar coordinates in reference triangle.
  • Figure 2: A qualitative comparison of the PDE solution $v_{T,R}$ with the volume integral in \ref{['Eqn_Approximation']}
  • Figure 3: Exponential convergence of $v_{T,R}$ for a periodic problem.
  • Figure 4: The absolute value of the Fourier transform of F
  • Figure 5: Exponential convergence of $v_{T,R}$ for a problem with vanishing frequency components in the origin ($\omega_0 = \sqrt{2}/8$).

Theorems & Definitions (21)

  • Lemma 1
  • Proof
  • Remark 1
  • Theorem 1
  • Proof : Proof of Theorem \ref{['Thm_Analysis_1']}
  • Theorem 2
  • Proof : Proof of Theorem \ref{['Thm_Analysis_b']}
  • Lemma 2
  • Proof : Proof of Lemma \ref{['Lemma_Per_ModError']}
  • Lemma 3
  • ...and 11 more