Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

TL;DR

This paper investigates the stability and cohomological properties of finite element spaces of differential forms across arbitrary dimensions, extending classical results and establishing new inequalities using algebraic topology tools.

Contribution

It introduces a framework linking algebraic topology with finite element spaces, extending exact sequence results to $hp$ spaces and proving discrete inequalities on manifolds.

Findings

Extension of exact sequence property to $hp$ finite element spaces

Development of regularization operators for differential forms

Proof of discrete Poincaré-Friedrichs and Rellich inequalities

Abstract

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological properties. The analysis applies in particular to certain $hp$ finite element spaces, extending results in trivial topology often referred to as the exact sequence property. Then we define regularization operators. Combined with the standard interpolators they enable us to prove discrete Poincar\'e-Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.