Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez
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引用次数: 0
Abstract
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或标量电势,两者各有利弊。在存在非三维拓扑的导体时,使用基于标量势的公式会出现僵局。补救办法是用第一同调群的生成器来扩展近似空间。现有的大多数算法都需要特殊的,如分层的有限元基础构造。利用花键的德拉姆复近似的见解,我们证明在这里不需要额外的条件。生成树技术可用于辅助控制网格上的微分形式样条空间和 de Rham 复数之间的同构所产生的六面体网格。
期刊介绍:
Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis.
This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.