High Order Methods with Exact Conservation Properties

Abstract

Theses notes present the duality of Whitney forms as a tool to describe manifolds by chains and fields by cochains. Relying on this duality, we can construct compatible discretization methods for PDEs, that are methods which respect the nature of the fields involved in the equations (the degrees of freedom have a physical meaning) as well as the geometric and topological structure of the continuous model. We briefly recall how it is possible to define high-order Whitney forms just refining the chains that describe the manifolds.