Extension Operators and Geometric Decompositions

Geometric decomposition is a widely used tool for constructing local bases for finite element spaces. For finite element spaces of differential forms on simplicial meshes, Arnold, Falk, and Winther showed that geometric decompositions can be constructed from extension operators satisfying certain properties. In this paper, we generalize their results to function spaces and meshes satisfying very minimal hypotheses, while at the same time reducing the conditions that must hold for the extension operators. In particular, the geometry of the mesh and the mesh elements can be completely arbitrary, and the function spaces need only have well-defined restrictions to subelements. In this general context, we show that extension operators yield geometric decompositions for both the function space and its dual. Later, we specialize to simplicial meshes, and we show that to obtain geometric decompositions, one needs only to construct extension operators on the reference simplex in each dimension. In particular, for simplicial meshes, the existence of geometric decompositions depends only on the dimension of the mesh.