Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms

TL;DR

This paper presents physics-compatible discretization methods on single and dual grids, comparing their effectiveness for solving the Poisson equation of volume forms, and clarifies the distinction between vectors, forms, and pseudo-forms.

Contribution

It introduces a dual grid and a single grid discretization framework based on differential forms, enhancing the understanding of physics-compatible numerical methods.

Findings

Dual grid method resembles staggered finite volume method

Single grid approach resembles finite element method

Both methods effectively solve the Poisson equation for volume forms

Abstract

This paper introduces the basic concepts for physics-compatible discretization techniques. The paper gives a clear distinction between vectors and forms. Based on the difference between forms and pseudo-forms and the $\star$-operator which switches between the two, a dual grid description and a single grid description are presented. The dual grid method resembles a staggered finite volume method, whereas the single grid approach shows a strong resemblance with a finite element method. Both approaches are compared for the Poisson equation for volume forms.