Finite element systems of differential forms
Provides a foundational framework for mixed finite element methods, unifying and extending existing approaches for polyhedral grids.
The paper develops a unified theory of mixed finite elements using inverse systems of differential form complexes over cellular complexes, enabling construction of commuting interpolators and smoothers with uniform stability in Lebesgue spaces. This yields eigenpair approximation for the Hodge-Laplacian and adapted Sobolev estimates.
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.