An Extended Galerkin analysis in finite element exterior calculus

TL;DR

This paper introduces new discontinuous Galerkin methods within the extended Galerkin framework for the Hodge--Laplace equation, providing a unified analysis and hybridization approach in finite element exterior calculus.

Contribution

It develops several families of DG methods in the extended Galerkin framework with a comprehensive inf-sup analysis and hybridization for the Hodge--Laplace equation.

Findings

Unified inf-sup stability analysis for all discretization parameters

Methods can be hybridized into a reduced two-field formulation

Applicable to contractible domains in finite element exterior calculus

Abstract

For the Hodge--Laplace equation in finite element exterior calculus, we introduce several families of discontinuous Galerkin methods in the extended Galerkin framework. For contractible domains, this framework utilizes seven fields and provides a unifying inf-sup analysis with respect to all discretization and penalty parameters. It is shown that the proposed methods can be hybridized as a reduced two-field formulation.