Finite element exterior calculus for parabolic problems

TL;DR

This paper extends finite element exterior calculus to parabolic problems, specifically the Hodge heat equation, analyzing both semidiscrete and fully-discrete Galerkin methods based on finite element differential forms.

Contribution

It introduces a novel extension of finite element exterior calculus to parabolic problems using mixed variational formulations and finite element differential forms.

Findings

Analysis of semidiscrete scheme stability

Analysis of fully-discrete scheme stability

Validation of method for Hodge heat equation

Abstract

In this paper, we consider the extension of the finite element exterior calculus from elliptic problems, in which the Hodge Laplacian is an appropriate model problem, to parabolic problems, for which we take the Hodge heat equation as our model problem. The numerical method we study is a Galerkin method based on a mixed variational formulation and using as subspaces the same spaces of finite element differential forms which are used for elliptic problems. We analyze both the semidiscrete and a fully-discrete numerical scheme.