Finite element systems of differential forms

TL;DR

This paper develops a comprehensive theory of mixed finite element systems for differential forms, enabling stable approximation of eigenvalues and Sobolev embeddings on complex polyhedral grids.

Contribution

It introduces a new framework using inverse complexes of differential forms, covering variable order polynomial elements and ensuring stability and commutation properties.

Findings

Stable eigenpair approximation for the Hodge-Laplacian.

Construction of commuting interpolators and smoothers.

Extension of Sobolev inequalities to discrete differential forms.

Abstract

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.