Approximation of fractional harmonic maps

TL;DR

This paper develops theoretical compactness results and numerical methods for approximating fractional harmonic maps, addressing nonlocality and unit-length constraints, with applications demonstrated in spin chain dynamics and defect modeling.

Contribution

It introduces new weak compactness results for fractional harmonic maps and analyzes numerical schemes ensuring their convergence.

Findings

Numerical methods successfully approximate fractional harmonic maps.

Convergence of numerical schemes is proven based on compactness results.

Numerical examples validate the effectiveness of the proposed methods.

Abstract

This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods.