Interpretation of a Discrete de Rham method as a Finite Element System

TL;DR

This paper reinterprets the Discrete de Rham (DDR) method as a finite element system with a computable discrete L2 product, enhancing analysis and connecting it with Virtual Element Methods.

Contribution

It provides a new perspective on DDR as a finite element system, strengthening conformity and consistency results without altering the original method.

Findings

Stronger conformity and consistency properties for DDR.

Unified analysis framework connecting DDR and FES.

Inclusion of Virtual Element Method in the discussion.

Abstract

We show that the DDR method can be interpreted as defining a computable consistent discrete $\mathrm{L}^2$ product on a conforming FES defined by PDEs. Without modifying the numerical method itself, this point of view provides an alternative approach to the analysis. The conformity and consistency properties we obtain are stronger than those previously shown, even in low dimensions. We can also recover some of the other results that have been proved about DDR, from those that have already been proved, in principle, in the general context of FES. We also bring VEM, the Virtual Element Method, into the discussion.