The Bubble Transform, Part Ii

The purpose of this note is to discuss a generalization of the bubble transform to differential forms. This transform was dicussed for ordinary functions, or zero-forms, in [4]. The present note represents the beginnig of a similar study for differential forms, where we build a transformation which decomposes any function in HΛk(Ω) into a sum of local bubbles supported on appropriate macroelements with respect to a given simplicial triangulation T of the domain Ω. A key points are that the transform in bounded in HΛk(Ω), and it should commute with the exterior derivative. Furthermore, the transform should preserve the piecewise polynomial structure of the standard finite element spaces of k-forms. The manuscript [4] gives a detailed construction and a rigorous analysis of the situation for zero-forms. In contrast, the present note is more restricted. However, when Ω ⊂ R2 the construction of the transform given below is close to complete.