A variational interpretation of the Biot–Savart operator and the helicity of a bounded domain
A variational interpretation of the Biot–Savart operator and the helicity of a bounded domain
复制标题
Biot-Savart算子的变分解释和有界域的螺旋度
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
A. Valli
中科院分区：
文献类型：
--
作者：
A. Valli
在这篇注记中，我们证明了Biot-Savart算子在与边界相切的无散度向量场空间上的投影是一个合适的鞍点变分问题的解。由于这个投影的Biot-Savart算子被证明是紧的，它的谱可以完全刻画。特别地，通过适当的有限元离散，通过确定具有最大绝对值的投影Biot-Savart算子的特征值，可以计算一般拓扑形状的有界域的螺旋度。在本文中，我们证明了Biot-Savart算子在与边界相切的无散度向量场空间上的投影是适当的鞍点变分问题的解。由于这个投影的Biot-Savart算子被证明是紧的，它的谱可以完全刻画。特别地，通过适当的有限元离散，通过确定具有最大绝对值的投影Biot-Savart算子的特征值，可以计算一般拓扑形状的有界域的螺旋度。
In this note, we show that the projection of the Biot–Savart operator over the space of divergence-free vector fields that are tangential to the boundary is the solution of a suitable saddle-point variational problem. Since this projected Biot–Savart operator is shown to be compact, its spectrum can be completely characterized. In particular, through a suitable finite element discretization, it becomes possible to compute the helicity of a bounded domain of a general topological shape, via the determination of the eigenvalue of the projected Biot–Savart operator that has a maximum absolute value.In this note, we show that the projection of the Biot–Savart operator over the space of divergence-free vector fields that are tangential to the boundary is the solution of a suitable saddle-point variational problem. Since this projected Biot–Savart operator is shown to be compact, its spectrum can be completely characterized. In particular, through a suitable finite element discretization, it becomes possible to compute the helicity of a bounded domain of a general topological shape, via the determination of the eigenvalue of the projected Biot–Savart operator that has a maximum absolute value.