Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

TL;DR

This paper introduces a fast, scalable finite element method for solving the magnetostatic scalar potential problem using a non-uniform FFT approach, enabling efficient computations for large-scale micromagnetic simulations.

Contribution

It develops a novel non-uniform FFT technique integrated with finite element methods to efficiently evaluate the scalar potential in micromagnetic problems.

Findings

Method scales as O(M + N + N log N) for large problems

Numerical tests confirm accuracy and efficiency

Generalizes convolution theorem for finite element context

Abstract

We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M + N + N log N) for N nodes and M surface triangles. We confirm our approach by several numerical tests.