A Construction of Spaces of Compatible Differential Forms on Cellular Complexes

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 devided into polyhedra, for which it provides for each $k$ a space of $k$-forms with a basis indexed by the set of $k$-dimensional polyhedra. 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. Furthermore the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh, can be obtained in this way, and the method provides generalizations to all dimensions. In the framework of mimetic finite differences the construction provides a conforming reconstruction operator.