Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

涡流问题的常见公式涉及矢量势或标量势，每种都有其自身的优点和缺点。当在存在具有非平凡拓扑的导体的情况下使用基于标量势的公式时，就会出现僵局。补救措施是用第一上同调群的生成元来扩充近似空间。大多数现有的算法需要特殊的，例如分层的、有限元基础构造。利用 de Rham 复数近似与样条的见解，我们表明附加条件在这里是不必要的。生成树技术可以适用于在由微分形式的样条空间和辅助控制网格上的 de Rham 复形之间的同构产生的六面体网格上进行操作。