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This paper introduces the basic concepts for physics-compatiblediscretization techniques. The paper gives a clear distinction between vectorsand forms. Based on the difference between forms and pseudo-forms and the$\star$-operator which switches between the two, a dual grid description and asingle grid description are presented. The dual grid method resembles astaggered finite volume method, whereas the single grid approach shows a strongresemblance with a finite element method. Both approaches are compared for thePoisson equation for volume forms.

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