# Discrete Poincaré Lemma

Author: Tania Maria Nunes Goncalves (University of Kent)

### Abstract

We know that the Poincaré lemma solves the equation $\omega=\mathrm{d}\eta$, if $\mathrm{d}\omega=0$, giving that $\eta=H(\omega)$, where $H$ is a homotopy operator. From the generalisation of the Poincaré lemma to the variational complex we can find the equivalence class of the conservation laws for a Lagrangian with a Lie symmetry. What do the homotopy operators of the de Rham complex and of the variational complex look like in a manifold $M$ with an open cover $\{U_1,...,U_n\}$? What do the conservation laws look like in such a space?

In this poster, we present the formulae for such homotopy operators. Then we show how these can be restricted to the simplicial complex and the differential complex of finite element spaces. With such homotopy operators we can thus obtain a finite dimensional approximation to a differential system with an analogous conservation law.