Extending geometric discretisation

Abstract

Geometric discretisation (GD) [1] is a novel approach capable of capturing topological properties, based on a correspondence between discrete objects and operations on a triangulation with continuum ones on a manifold. We felt that much work remained to be done to fullfill the potential it appeared to have and we were not wrong. We began by trying to incorporate metric into the scheme where we found that a cubic formulation, which required the introduction of a non-trivial variation of the Whitney map [2], was well suited to the task. These were our initial goals. Along the way we found an interesting space which is generated by the Whitney map. This is a finite dimensional space (FDS) which means that operators can be expressed as matrices and provides another discretisation scheme altogether; a variation of finite element methods in some sense. With this for example our discrete wedge product is associative which is not the case in GD. Unfortunately topology is lost in this process though.