An Extended Galerkin Analysis for Elliptic Problems

TL;DR

This paper introduces a comprehensive Galerkin analysis framework for elliptic problems that unifies and extends the understanding of various finite element and discontinuous Galerkin methods through a four-variable discretization approach.

Contribution

It develops a general analysis framework employing four discretization variables, proving uniform inf-sup conditions, and unifying the analysis of many existing finite element methods.

Findings

Most finite element and DG methods fit into this framework.

The framework ensures stability through uniform inf-sup conditions.

It allows analysis of methods with different discretization and penalization parameters.

Abstract

A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $u_h, \bm{p}_h, \check u_h$ and $\check p_h$, where $u_h$ and $\bm{p}_h$ are for approximation of $u$ and $\bm{p}=-\alpha \nabla u$ inside each element, and $ \check u_h$ and $\check p_h$ are for approximation of residual of $u$ and $\bm{p} \cdot \bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.