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.

中文翻译：

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分数谐波映射的近似

本文讨论了分数谐波映射的近似。除了单位长度约束外，还必须解决非定域性的困难。我们建立了单位长度向量场上分数狄利克雷能量临界点的弱紧致性结果。我们设计并分析了数值方法，用于逼近与分数谐波映射相关的各种偏微分方程。紧致性结果意味着数值近似的收敛。给出了自旋链动力学和点缺陷的数值例子来证明所提出方法的有效性。