Differential complexes and numerical stability

TL;DR

This paper discusses how differential complexes, like the de Rham complex, are crucial in designing stable numerical methods for PDEs by preserving geometric structures, leading to new stable discretizations.

Contribution

It highlights the importance of discrete differential complexes in ensuring stability and structure preservation in numerical PDE methods, offering a unifying geometric framework.

Findings

Differential complexes are essential for stable PDE discretizations.

Discrete complexes help preserve geometric structure in numerical methods.

The approach unifies and advances the design of numerical PDE schemes.

Abstract

Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.