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

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 ⋆-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. By defining a suitably weighted inner product for 1-forms this approach can readily be applied to anisotropic diffusion models for volume forms.