Eulerian and Semi-Lagrangian Methods for Convection-Diffusion for Differential Forms

Abstract

We consider generalized linear transient convection-diffusion problems for differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie derivatives with respect to a prescribed smooth vector field. We construct both new Eulerian and semi-Lagrangian approaches to the discretization of the Lie derivatives in the context of a Galerkin approximation based on discrete differential forms. Details of implementation are discussed as well as an application to the discretization of eddy current equations in moving media.