AI 中文总结
本文针对阿贝尔结构群非平凡主丛上的杨-米尔斯联络，提出有限元方法，通过拉格朗日乘子弱强制跳跃条件，证明鞍点问题适定性并推导收敛估计，霍普夫纤维化实验验证其线性收敛性。
AI 中文摘要
针对具有阿贝尔结构群的非平凡主丛上的杨-米尔斯联络，我们提出一种有限元方法。局部联络形式满足由主丛转移函数诱导的内部跳跃条件。我们在间断有限元外微分演算空间中离散这些形式，并通过拉格朗日乘子弱强制跳跃条件。我们证明所得鞍点问题适定，并推导出先验收敛估计。针对霍普夫纤维化的数值实验展示了该方法，且当球面被分段线性网格近似时呈现线性收敛。
英文摘要
We propose a finite element method for Yang--Mills connections on nontrivial principal bundles with abelian structure group. Local connection forms satisfy internal jump conditions induced by the transition functions of the bundle. We discretize these forms in broken finite element exterior calculus spaces and enforce the jump conditions weakly by Lagrange multipliers. We prove well-posedness of the resulting saddle-point problem and derive an a priori convergence estimate. A numerical experiment for the Hopf fibration illustrates the method and exhibits linear convergence when the sphere is approximated by a piecewise linear mesh.
Comments38 pages
