A construction of spaces of compatible differential forms on cellular complexes
A construction of spaces of compatible differential forms on cellular complexes
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DOI：
10.1142/s021820250800284x
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发表时间：
2008-05-01
影响因子：
3.5
通讯作者：
Christiansen, Snorre H.
中科院分区：
文献类型：
--
作者：
Christiansen, Snorre H.
给定一个胞腔复形，我们构造微分形式空间，它在外导数下形成一个复形，这个复形同构于胞腔复形的上链复形。该构造特别适用于划分为多面体的欧几里得空间的子集，对于每个k，它提供了具有由k维单元集合索引的基的k形式的空间。在模拟有限差分的框架下，该构造提供了一个相容的重构算子。这种构造需要在每个胞元上具有微分形式的辅助空间，为此我们提供了两个例子。当单元是单形时，该构造可用于恢复标准混合有限元空间，也称为Whitney形式。我们也可以恢复先前由A. Buffa和作者讨论了二维网格的重心细化。
Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.