Face Bubble Functions in Finite Element Methods

• Face bubble functions are local enrichment objects attached to codimension-1 simplices, providing essential trace preservation in finite element exterior calculus.
• They decompose fields via the bubble transform using barycentric maps, cutoff factors, and Whitney forms to separate global and local contributions.
• These functions play a crucial role in multiscale methods by distinguishing face-related enrichments from interior bubbles to improve stability in perforated domains.

Searching arXiv for recent and foundational papers on face bubble functions and related bubble-transform / finite-element bubble enrichments. Face bubble functions are local enrichment objects attached to codimension-1 simplices of a mesh—edges in two dimensions and faces in three dimensions—and are most naturally understood within simplicial finite element and finite element exterior calculus (FEEC) settings. In the literature considered here, the term does not designate a single universally fixed construction. In one important line of work, a face bubble is the codimension-1 contribution in a simplicial bubble decomposition supported on the macroelement around a face; in another, apparently similar constructions combine edge-associated nonconforming basis functions with distinct element-interior bubbles and therefore should not be described literally as face bubbles. The resulting distinction between face-supported, edge-associated, and element-interior enrichment is central to the subject (Falk et al., 2023, Degond et al., 2013).

1. Terminology and scope

A useful starting point is the geometric indexing of local contributions by simplices. For a simplicial triangulation $\mathcal T$ of a domain $\Omega$, the bubble transform decomposes a piecewise smooth differential form $u\in \mathcal A^k(\mathcal T)$ as

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$

Here $W^k u$ is a global trimmed piecewise linear $k$-form, while each $B_f^k u$ is a local bubble associated with a simplex $f$; when $f\in \Delta_{n-1}(\mathcal T)$, the term is the face-associated layer of the decomposition (Falk et al., 2023).

The most direct finite-element description of a classical face bubble, as contrasted in the multiscale literature, is a function that is associated with one face or edge, vanishes on the rest of the element boundary, and is often supported on one or two neighboring elements sharing that face. This description matters because some methods that are strongly face-related are not, strictly speaking, face-bubble methods. In particular, the Crouzeix–Raviart multiscale construction in perforated media uses edge-associated basis functions $\Phi_e$ together with separate element-supported bubbles $\Omega$0; the former are face-related in the nonconforming sense, whereas the latter are interior bubbles (Degond et al., 2013).

Construction	Face-related object	Bubble object
Classical face bubble picture	Codimension-1 simplex contribution	Same local object
CR MsFEM in perforated media	Edge-associated $\Omega$1	Element-interior $\Omega$2
Bubble transform	$\Omega$3 for $\Omega$4	Part of an all-simplices hierarchy

This terminological separation prevents a common misunderstanding: not every method that exploits edge averages, face moments, or interelement traces is using face bubbles in the strict sense. Some methods are better described as face- or edge-based skeleton constructions augmented by interior bubbles (Degond et al., 2013).

2. Geometric and functional structure

The canonical support region for a simplex-indexed bubble is the macroelement, or star,

$\Omega$5

For a codimension-1 simplex $\Omega$6, this is exactly the face patch consisting of the cells that contain that face. The associated local target space is $\Omega$7, the piecewise smooth $\Omega$8-forms supported on $\Omega$9; in the polynomial setting, the relevant local spaces are $u\in \mathcal A^k(\mathcal T)$0 and $u\in \mathcal A^k(\mathcal T)$1 (Falk et al., 2023).

The bubble transform is built from barycentric maps, cutoff factors, Whitney forms, averaging operators, and trace-preserving operators. For a simplex $u\in \mathcal A^k(\mathcal T)$2, the barycentric map is

$u\in \mathcal A^k(\mathcal T)$3

and a key local factor is

$u\in \mathcal A^k(\mathcal T)$4

For codimension-1 simplices, $u\in \mathcal A^k(\mathcal T)$5 plays the role of a barycentric distance-to-face factor and is closely related to classical face-bubble factors. The construction also uses Whitney forms $u\in \mathcal A^k(\mathcal T)$6, which provide canonical lowest-order edge-, face-, and cell-type geometric basis objects (Falk et al., 2023).

A major development of the modern theory is the requirement that each local bubble preserve the regularity class of the original field individually. Earlier decompositions had locality and commutation properties, but the separate local bubbles were generally rational functions and space preservation appeared only after summing groups of local terms. The improved decomposition is strictly space-preserving: if $u\in \mathcal A^k(\mathcal T)$7 is piecewise smooth, then each $u\in \mathcal A^k(\mathcal T)$8 is piecewise smooth; if $u\in \mathcal A^k(\mathcal T)$9 belongs to a standard polynomial FEEC space, then each $$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$0 remains in the corresponding local polynomial space (Falk et al., 2023).

3. Face bubbles, edge moments, and nonconforming multiscale bases

A particularly instructive case is the Crouzeix–Raviart MsFEM for diffusion and advection–diffusion in perforated media. The perforated domain is

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$1

and the model problem in dimension $$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$2 is

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$3

The multiscale space is nonconforming and weakly continuous across mesh edges through the condition

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$4

This is the precise locus at which the method becomes face- or edge-based: continuity is enforced in edge average, not pointwise (Degond et al., 2013).

The local edge-associated basis functions $$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$5 on a coarse element $$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$6 solve

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$7

with edge-average constraints

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$8

and constant-flux conditions

$$u = W^k u + \sum_{f\in \Delta(\mathcal T)} B_f^k u = W^k u + \sum_{m=0}^n \sum_{f\in \Delta_m(\mathcal T)} B_f^k u.$$9

In two dimensions these are edge-based objects; in three dimensions they would be the face-associated part of the construction (Degond et al., 2013).

The enrichment functions called bubbles in that paper are different. For each coarse element $W^k u$0, the bubble $W^k u$1 solves

$W^k u$2

with

$W^k u$3

These are classical element-interior bubbles, not face bubbles. The global approximation is written as a sum of edge-associated CR basis functions and element bubbles,

$W^k u$4

The paper therefore combines a face- or edge-based skeleton component with a separate interior enrichment, rather than defining bubbles on faces themselves (Degond et al., 2013).

4. Approximation roles and stability mechanisms

The multiscale perforated-media setting clarifies why one might want both face-related functions and bubbles. The CR edge basis is designed to reduce sensitivity to perforations intersecting coarse-grid boundaries, because it imposes integral constraints and constant normal fluxes rather than artificial linear Dirichlet traces along coarse edges. The element-interior bubbles are introduced for a different reason: with very dense perforations, standard basis functions may contribute insignificantly in the interior of a coarse element, and the bubble enrichment remedies that interior approximation deficit (Degond et al., 2013).

The numerical results make the separation of roles explicit. In a non-intersecting perforation configuration on an $W^k u$5 coarse mesh, relative $W^k u$6 errors are $W^k u$7 for CR MsFEM with bubbles and $W^k u$8 for standard MsFEM with bubbles. When perforations are shifted so that all coarse edges coincide with perforations, the corresponding errors become $W^k u$9 and $k$0. For the advection–diffusion test with $k$1 and $k$2, the $k$3 coarse-grid results are $k$4 for CR MsFEM with bubbles and $k$5 for standard MsFEM with bubbles. These data show that bubble enrichment alone is not sufficient when the dominant error source is the coarse-edge boundary treatment; the CR edge formulation addresses the face-related part of the difficulty, and the interior bubbles improve local resolution inside coarse blocks (Degond et al., 2013).

The Stokes literature provides a complementary caution. In the stabilized mixed finite element analysis of the Stokes problem, the bubble functions studied are element-interior functions $k$6 that vanish on the full element boundary, with fine-scale velocities of the form

$k$7

The paper shows that standard interior bubble enrichment works well for T3 and TET4 but that a direct equivalence between subgrid-based stabilized methods and Galerkin methods enriched by bubble functions cannot be constructed for Q4 and B8 using standard bubble functions; enriched Q4 retains the checkerboard mode. No face bubble construction is proposed there, but the result sharply delineates the limitations of interior bubbles and suggests why richer codimension-1 enrichments may be of interest (0806.3099).

5. Bubble transforms and the simplex hierarchy

Within FEEC, face bubble functions appear not as isolated ad hoc enrichments but as one layer of a mesh-geometric decomposition indexed by all simplices. The local operators $k$8 are defined so that every simplex contributes one bubble term, including codimension-1 faces. For $k$9, $B_f^k u$0,

$B_f^k u$1

while for cells $B_f^k u$2,

$B_f^k u$3

Thus a face bubble is the contribution associated with $B_f^k u$4, supported in the face patch $B_f^k u$5 (Falk et al., 2023).

This hierarchy is governed by trace preservation. The operators $B_f^k u$6 satisfy

$B_f^k u$7

and the decomposition follows from the telescoping identity

$B_f^k u$8

The interpretation is recursive: vertex traces, then edge traces, then face traces, then cell-interior residuals are peeled off in sequence. Face bubbles are therefore the codimension-1 part that remains after incompatible higher- and lower-dimensional trace information has been separated (Falk et al., 2023).

The scalar case $B_f^k u$9 makes the picture particularly transparent. On a triangle, the decomposition consists of vertex bubbles, edge bubbles, and cell bubbles. On a tetrahedron, it consists of vertices, edges, triangular faces, and tetrahedron interiors. In three dimensions, a face bubble attached to a triangle $f$0 is supported on the two tetrahedra sharing that triangle, or on one tetrahedron at the boundary; this is exactly the support pattern emphasized as the practical 3D face-bubble configuration (Falk et al., 2023).

6. Invariance, commuting structure, and terminological disambiguation

The theoretical strength of the space-preserving bubble transform lies in four properties. First, it is local:

$f$1

so $f$2. Second, it preserves standard FEEC polynomial spaces simplex-by-simplex:

$f$3

Third, it commutes with the exterior derivative,

$f$4

Fourth, it satisfies the stable decomposition estimate

$f$5

These properties are precisely what make face-local contributions usable in de Rham-compatible discretizations, Schwarz methods, subspace correction, local solvers, and multilevel constructions (Falk et al., 2023).

The phrase “face bubble functions” also appears in contexts that are only partially related or entirely unrelated to finite element face enrichments. In computer vision, “EmFace” represents a grayscale face image as a finite sum of weighted anisotropic Gaussian components over the continuous image domain,

$f$6

and is conceptually close to a superposition of localized blobs, but the terms are not compactly supported, not mesh-attached, and not a classical face-bubble basis (Zhang et al., 2021). In quantum field theory, “bubble wall” refers to the interface between symmetry-restored and symmetry-broken phases, and “Green’s functions in the presence of a bubble wall” concern spectral representations of propagators with spatially varying masses; this usage of “bubble” is physically unrelated to finite-element bubble functions (Kubota, 2024).

Taken together, these literatures support a precise contemporary interpretation. In the strict finite-element sense, face bubble functions are codimension-1 local contributions supported on face patches and organized by trace, support, and polynomial-invariance requirements. In neighboring multiscale and stabilized formulations, one often encounters edge-associated skeleton functions plus interior bubbles rather than literal face bubbles. The distinction is not terminological trivia: it determines which degrees of freedom control interelement transmission, which enrich the element interior, and which theoretical properties—trace preservation, de Rham compatibility, or mixed stability—can be expected from the construction (Falk et al., 2023, Degond et al., 2013, 0806.3099).