Extension operators and geometric decompositions
Yakov Berchenko-Kogan
TL;DR
The paper addresses constructing local bases for finite element spaces on general meshes by extending geometric decompositions beyond the original differential-form setting. It introduces a poset-based framework for function spaces with trace maps, proving that local extension operators on vanishing-trace subspaces suffice to produce geometric and dual decompositions, and that in the simplicial case the existence of geometric decompositions reduces to constructing extension operators on the reference simplex in each dimension, i.e., the problem depends only on the mesh dimension. Key results include the geometric decomposition theorem $\mathcal{F}(\mathcal{T}) = \bigoplus_{F\in\mathfrak T} E_F^{\mathcal{T}} \mathring{\mathcal{F}}(F)$ and the dual decomposition $\mathcal{F}(\mathcal{T})^* = \bigoplus_{F\in\mathfrak T} (\mathrm{tr}^{\mathcal{T}}_F)^* \mathring{\mathcal{F}}(F)^{\dagger}$, with a simplicial-extension theorem tying these to extensions on reference simplices $T^n$ and $Q^{n+1}$. The framework applies to broader spaces, including covariant tensors, and simplifies the construction of local bases by reducing the problem to dimension-based extension on reference elements. This has practical implications for building efficient FE bases on diverse mesh types and for spaces beyond differential forms.
Abstract
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 primal and dual function spaces. 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.