The Bubble Transform and the de Rham Complex

The purpose of this paper is to discuss a generalization the bubble transform differential forms. was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, represents new tool understanding finite element spaces arbitrary polynomial degree. present contains similar study From simplicial mesh $${{\mathscr {T}}}$$ domain $$\varOmega $$ , we build map which decomposes piecewise smooth k-forms into sum local bubbles supported on appropriate macroelements. key properties decomposition are that it commutes with exterior derivative preserves structure standard k-forms. Furthermore, bounded $$L^2$$ also subspace consisting derivatives .