In this paper, we explore a new tool for analyzing finite element methods called the bubble transform. This technique allows us to decompose differential forms into simpler pieces called bubbles, which have local support on domains defined by a given simplicial mesh of the domain. The bubble transform has several useful properties, such as commuting with the exterior derivative and preserving certain polynomial structures. Our goal is to refine the earlier theory developed in [6] to make it more comprehensive and accurate.
Section 2: Assumptions and Notation
Before diving into the construction of our decomposition, we first list some assumptions we will make and recall some standard notation and properties of differential forms and simplicial complexes. We also define some basic operators that we will use in the construction of our decomposition.
Section 3: Scalar-Valued Functions
To motivate the general theory, we discuss the decomposition of scalar-valued functions in Section 3. This helps us understand how the bubble transform works and its potential applications.
Section 4: Local Functions and Construction
In Section 4, we define various local functions depending on the given mesh T . The recursive construction of these mesh functions is a new approach compared to [6] and plays a crucial role in obtaining improved results. We use these local functions to define families of order reduction operators that can simplify complex calculations.
Summary
The bubble transform is a powerful tool for analyzing finite element methods, allowing us to decompose differential forms into simpler pieces called bubbles with local support on domains defined by a given simplicial mesh. This technique has several useful properties, such as commuting with the exterior derivative and preserving certain polynomial structures. By refining the earlier theory and developing new construction methods, we can improve our understanding of the bubble transform and its applications in mathematics and computer science.
