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Local space-preserving decompositions for the bubble transform

Richard S. Falk, Ragnar Winther

TL;DR

The authors develop a strictly space-preserving bubble transform for differential $k$-forms on a triangulated domain, yielding a decomposition $u = W^k u + 1_{f} B_f^k u$ in which each local bubble $B_f^k u$ preserves piecewise smoothness and respects standard FEEC spaces. The construction builds a telescope of operators $C_m^k$, introduces local bubbles $K_{m,f}^k$ and order-reduction operators $R_{e,f}^k$ guided by carefully designed weight functions $z_{e,f}$ and mesh-dependent functions $eta$, $eta_e$, ensuring $d$-commutation and $L^2$-boundedness. A key innovation is the representation of $C_m^k - C_{m-1}^k$ as sums of local operators $K_{m,f}^k$, which leads to a fundamental identity that enables a decomposition into local bubbles with controlled support and invariance properties; all operators preserve piecewise polynomial spaces and commute with $d$. The resulting stable, space-preserving decomposition applies uniformly across dimensions and FE spaces, offering hp-FEM tools and robust preconditioners for Hodge-Laplace problems without relying on extension operators. Overall, the work provides a rigorous, mesh-dependent framework for localizing differential-form decompositions with strong theoretical guarantees and practical implications for hp-adaptive finite element methods.

Abstract

The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L^2, and satisfies the stable decomposition property.

Local space-preserving decompositions for the bubble transform

TL;DR

The authors develop a strictly space-preserving bubble transform for differential -forms on a triangulated domain, yielding a decomposition in which each local bubble preserves piecewise smoothness and respects standard FEEC spaces. The construction builds a telescope of operators , introduces local bubbles and order-reduction operators guided by carefully designed weight functions and mesh-dependent functions , , ensuring -commutation and -boundedness. A key innovation is the representation of as sums of local operators , which leads to a fundamental identity that enables a decomposition into local bubbles with controlled support and invariance properties; all operators preserve piecewise polynomial spaces and commute with . The resulting stable, space-preserving decomposition applies uniformly across dimensions and FE spaces, offering hp-FEM tools and robust preconditioners for Hodge-Laplace problems without relying on extension operators. Overall, the work provides a rigorous, mesh-dependent framework for localizing differential-form decompositions with strong theoretical guarantees and practical implications for hp-adaptive finite element methods.

Abstract

The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L^2, and satisfies the stable decomposition property.
Paper Structure (16 sections, 23 theorems, 192 equations, 5 figures)

This paper contains 16 sections, 23 theorems, 192 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Assumptions
  4. Notation
  5. The macroelements
  6. The average operators and their generalizations
  7. An outline of the construction
  8. The case of scalar-valued functions
  9. The local structure of the mesh
  10. The functions $\mu_e(f)$
  11. Construction of the weight functions $z_{e,f}$
  12. The order reduction operators
  13. The local operators $K_{m,f}^k$
  14. Verifying the fundamental identity
  15. Bounding the operator norms
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $f \in \Delta_{m-1}({\mathcal{T}})$ and $0 \le k < n-m$. If $c \in {\mathscr C}_k(f^*)$ satisfies $\partial_k(f^*) c = 0$, then there is a $\tilde{c} \in {\mathscr C}_{k+1}(f^*)$ such that $c = \partial_{k+1}(f^*) \tilde{c}$. Furthermore, $\tilde{c}$ is uniquely determined if we require $\delta_

Figures (5)

  • Figure 1: The map $x \mapsto L_f(x)$ and $y \mapsto L_f(y)$ for $n=2$, $m=1$, and $f=[x_0,x_1]$.
  • Figure 2: 2D Vertex macroelement 2D Edge macroelement When $f$ is a vertex, $f^*$ is the boundary of the vertex macroelement. When $f$ is an edge, $f^*$ is the two remaining vertices of the macroelement.
  • Figure 3: The vertex macroelement $\Omega_f \subset \mathbb{R}^3$, where $f = [x_0]$ and $f^*$ is the union of the faces containing three of the other vertices (excluding $x_0$)
  • Figure 4: The edge macroelement $\Omega_f \subset \mathbb{R}^3$, where $f = [x_0,x_1]$ and $f^*$ is the closed curve connecting the vertices $x_2,x_3,x_4$, and $x_5$.
  • Figure 5: The face macroelement $\Omega_f \subset \mathbb{R}^3$, where $f = [x_4,,x_2,x_1]$ and $f^*$ is the vertices $x_3$, and $x_5$.

Theorems & Definitions (43)

  • Lemma 2.1
  • Remark 1
  • Remark 2
  • Lemma 2.2
  • Remark 3
  • Remark 4
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 4.1
  • proof
  • ...and 33 more