Curl-Based Pushforward Methods

• Curl-Based Pushforward is a construction that uses the curl operator to map scalar and vector fields in an exact, norm-preserving manner in both continuous and discrete settings.
• It enables effective reconstruction in computational electromagnetics by applying constrained lifting and compatibility in finite element methods.
• Recent developments extend the concept to measure-theoretic and div-curl stability frameworks, offering rigorous tools for nonlinear differential analysis.

Curl-based pushforward denotes a family of structure-preserving constructions in which passage from one object to another is controlled by the curl operator, by exact-sequence compatibility, or by the chain/cochain analogue of exterior differentiation. In the current literature, this includes an exact 2D map from a scalar curl-curl problem to a vector curl-curl problem via $\mathbf{E}=\mathbf{curl}\,F$ and its discrete realization (Gerritsma et al., 2018), patchwise $H(\mathrm{curl})$-constrained liftings used in equilibrated reconstruction (Chaumont-Frelet et al., 2021), curl-compatible finite element complexes on cubical meshes without explicit Piola formulas (Zhang et al., 2022), and Sobolev pushforwards of currents satisfying $\partial(f_\#T)=f_\#(\partial T)$, where coboundary plays the role of a weak exterior derivative (Ikonen, 2023). By contrast, several recent uses of “pushforward” are explicitly unrelated to curl: approximate pushforward designs are purely measure-/moment-theoretic (Czartowski et al., 1 Dec 2025), and pushforward monads are categorical right Kan extensions (Mateo, 2024). This suggests that “curl-based pushforward” is best understood as a technically specific, context-dependent notion rather than a single standardized construction.

1. Terminological scope and exclusions

The literature separates sharply between curl-mediated transport and other notions of pushforward. In approximate quantum pushforward design theory, the pushforward is the ordinary measure-theoretic direct image

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$

and the analysis concerns moment operators, Schatten norms, and Lipschitz bounds for induced maps on $t$-copy moments. That framework explicitly states that there is no connection to geometric pushforwards of vector fields, differential forms, Jacobians, divergence, or curl; “pushforward” there is entirely design-theoretic and measure-theoretic (Czartowski et al., 1 Dec 2025).

An analogous separation appears in category theory. Pushforward monads define

$G_\#T:=\operatorname{Ran}_G GT,$

a monad obtained by right Kan extension along a functor $G$. This construction is universal and algebraic, with no discussion of curl, divergence, vector calculus, or geometric transport (Mateo, 2024).

Within computational electromagnetics and compatible discretization, the relevant objects are different. One paper develops a local curl-preserving reconstruction operator into $H(\mathrm{curl})$-conforming Nédélec spaces and states explicitly that this is not a geometric pushforward induced by a map between physical domains, but rather a local constrained lifting/projection into $H(\mathrm{curl})$ (Chaumont-Frelet et al., 2021). Another constructs cubical finite element families whose compatibility is curl-based, yet also does not present an explicit reference-to-physical pushforward formula (Zhang et al., 2022). The terminology is therefore best reserved for settings where transport is mediated by curl itself, or by a weak chain/cochain formalism encoding the same differential structure.

2. Exact curl-induced transfer in equivalent curl-curl problems

A literal curl-based pushforward appears in the two-dimensional equivalence between a scalar unknown $F$ and a vector field $H(\mathrm{curl})$0. The scalar curl and vector curl are defined by

$H(\mathrm{curl})$1

and the continuous relation is

$H(\mathrm{curl})$2

With matching boundary data, the paper states that the vector curl-curl problem and the scalar curl-curl problem are equivalent in the sense that $H(\mathrm{curl})$3 solves the vector problem and $H(\mathrm{curl})$4 solves the scalar problem iff $H(\mathrm{curl})$5 (Gerritsma et al., 2018).

The distinctive feature is that this identity is preserved exactly at the discrete level. The scalar unknown $H(\mathrm{curl})$6 is represented in a primal nodal basis, while the vector field $H(\mathrm{curl})$7 is represented in dual degrees of freedom. The discrete curl is encoded by the incidence matrix $H(\mathrm{curl})$8, and the Riesz map to dual edge coefficients is $H(\mathrm{curl})$9. The coefficient-level relation is

$\partial(f_\#T)=f_\#(\partial T)$0

which is the exact algebraic analogue of $\partial(f_\#T)=f_\#(\partial T)$1. The weak adjoint structure is likewise exact: the curl of the dual vector field is represented by $\partial(f_\#T)=f_\#(\partial T)$2 together with a boundary restriction matrix $\partial(f_\#T)=f_\#(\partial T)$3. As a result, the discrete scalar and vector systems map into one another without approximation error.

The paper also proves equality of the discrete energy norms,

$\partial(f_\#T)=f_\#(\partial T)$4

so the pushforward by curl is norm-preserving in the natural discrete setting. Numerically, the difference between the computed vector field and $\partial(f_\#T)=f_\#(\partial T)$5 is reported as $\partial(f_\#T)=f_\#(\partial T)$6, the norms are exactly equal for the displayed polynomial degrees, and both converge exponentially to the exact norm

$\partial(f_\#T)=f_\#(\partial T)$7

In this formulation, curl-based pushforward is not an analogy: it is an exact continuous and discrete transfer principle.

3. Curl-preserving local lifting into $\partial(f_\#T)=f_\#(\partial T)$8

A different meaning arises in equilibrated a posteriori analysis for the curl-curl problem. Here the starting point is a discrete magnetic vector potential

$\partial(f_\#T)=f_\#(\partial T)$9

satisfying the lowest-order orthogonality condition

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$0

Since $F_*(\mu)[B]:=\mu[F^{-1}(B)],$1 is only piecewise polynomial and generally not $F_*(\mu)[B]:=\mu[F^{-1}(B)],$2-conforming, it cannot itself serve as an equilibrated flux. The reconstruction therefore seeks

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$3

whose curl matches the physical current $F_*(\mu)[B]:=\mu[F^{-1}(B)],$4 exactly in the polynomial-data case, or up to controlled oscillation otherwise (Chaumont-Frelet et al., 2021).

The construction is patchwise. At the continuous level one defines

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$5

with

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$6

so that

$F_*(\mu)[B]:=\mu[F^{-1}(B)],$7

The discrete method reproduces this through a multistage local procedure: an over-constrained Raviart–Thomas minimization for $F_*(\mu)[B]:=\mu[F^{-1}(B)],$8, an elementwise divergence-free decomposition $F_*(\mu)[B]:=\mu[F^{-1}(B)],$9, formation of patch currents

$t$0

a divergence-free polynomial projection $t$1, and finally the Nédélec patch reconstruction

$t$2

When $t$3, the reconstruction satisfies

$t$4

The central stability statement is $t$5-robust best-approximation-type control: $t$6 with a constant independent of $t$7. In this setting, the closest precise meaning of curl-based pushforward is a curl-preserving local lifting or constrained $t$8-projection into an $t$9-conforming affine space. The paper itself emphasizes that this is not a geometric pushforward induced by a map $G_\#T:=\operatorname{Ran}_G GT,$0.

4. Curl-compatible transfer on cubical finite element complexes

On cubical meshes, the pushforward language is again implicit rather than explicit. The finite element construction is carried out directly on each physical cube

$G_\#T:=\operatorname{Ran}_G GT,$1

and the paper does not introduce a reference cube, a map $G_\#T:=\operatorname{Ran}_G GT,$2, or standard Piola formulas. Nevertheless, the resulting local and global spaces are organized entirely around curl-compatibility (Zhang et al., 2022).

For $G_\#T:=\operatorname{Ran}_G GT,$3, the enriched local $G_\#T:=\operatorname{Ran}_G GT,$4-type space is

$G_\#T:=\operatorname{Ran}_G GT,$5

while the enriched $G_\#T:=\operatorname{Ran}_G GT,$6-type space is

$G_\#T:=\operatorname{Ran}_G GT,$7

That identity is the key structural statement: the enrichment on the $G_\#T:=\operatorname{Ran}_G GT,$8 side is exactly the curl-image of the enrichment on the $G_\#T:=\operatorname{Ran}_G GT,$9 side.

The new nonconforming regularity is encoded by moments of the face tangential trace of the curl,

$G$0

and the global continuity condition is

$G$1

At the bubble level, if

$G$2

then

$G$3

which is the mechanism by which the additional face continuity is enforced and the new bubble coefficients are identified.

The commuting structure is explicit: $G$4 These identities are embedded in the exact nonconforming finite element Stokes complex

$G$5

The resulting picture is pushforward-equivalent rather than pushforward-explicit. The spaces, degrees of freedom, and commuting interpolators preserve exactly the trace and operator information that a covariant/contravariant pushforward would normally be designed to preserve, but the paper encodes this directly in physical-element definitions rather than through a named pushforward map.

5. Currents, cochains, and the weak differential analogue

A broader geometric-measure-theoretic realization of curl-based pushforward appears in the pushforward of currents under Sobolev maps. For a Riemannian manifold $G$6, a complete metric space $G$7, and

$G$8

there exist a Borel representative $G$9 of $H(\mathrm{curl})$0, a Borel $H(\mathrm{curl})$1, and a chain morphism

$H(\mathrm{curl})$2

such that for $H(\mathrm{curl})$3 and Borel $H(\mathrm{curl})$4,

$H(\mathrm{curl})$5

The pushforward is defined only outside a $H(\mathrm{curl})$6-exceptional family of currents; “almost every” is understood in this modulus sense (Ikonen, 2023).

The construction replaces the classical Lipschitz formula by a Sobolev approximation scheme. The map $H(\mathrm{curl})$7 is approximated by Lipschitz maps into an injective Banach space, a Lusin–Lipschitz decomposition $H(\mathrm{curl})$8 is built, and the pushforward is assembled piecewise: $H(\mathrm{curl})$9 The decisive structural identity is

$H(\mathrm{curl})$0

so $H(\mathrm{curl})$1 is a chain morphism. This is the chain-level Stokes compatibility that replaces pointwise differential formulas in nonsmooth metric-target settings.

The dual objects are cochains. For a cochain $H(\mathrm{curl})$2, the pullback is defined by

$H(\mathrm{curl})$3

and the coboundary is

$H(\mathrm{curl})$4

For smooth bounded forms $H(\mathrm{curl})$5, the induced cochain $H(\mathrm{curl})$6 satisfies

$H(\mathrm{curl})$7

This is the precise weak analogue of compatibility with exterior differentiation. In a curl-oriented interpretation, boundary/coboundary takes over the role played by curl in classical vector calculus: closed cochains correspond to weakly closed differential data, and the pushforward preserves the chain complex needed for Stokes-type identities. The paper further uses this machinery to derive isoperimetric inequalities for Sobolev mappings and to prove sharp $H(\mathrm{curl})$8-Hölder continuity for quasiregular $H(\mathrm{curl})$9-curves.

6. Div-curl stability, Jacobians, and transported nonlinear quantities

The compensated-compactness perspective provides a final meaning for curl-based pushforward: weak stability of nonlinear quantities whose algebraic structure couples divergence-free and curl-free fields. The central div-curl theorem considers sequences

$F$0

under the improved exponent condition

$F$1

together with weak convergence, compactness of $F$2 and $F$3, and the assumption

$F$4

When the limit pair is regular enough, the conclusion is

$F$5

at the critical relation

$F$6

additional equi-integrability is required (Briane et al., 2015).

The proof exploits compactness on spherical slices. The key embedding is

$F$7

and the compactness argument proceeds by selecting annuli on which the spherical gradients are controlled, in the spirit of De Giorgi and Manfredi. This produces strong convergence on many spheres even when global compactness is unavailable. The necessity of the extra assumptions at the critical threshold is confirmed by counterexamples.

The main application to geometric transport is the Jacobian. Defining

$F$8

one has

$F$9

The Jacobian is therefore a div-curl product, and the theorem yields weak continuity results for $H(\mathrm{curl})$00 under $H(\mathrm{curl})$01 convergence with $H(\mathrm{curl})$02, and also in the critical $H(\mathrm{curl})$03 regime under Lorentz-space equi-integrability. The sharpness examples show that without the extra equi-integrability one can obtain concentration measures instead of the expected weak limit.

This suggests a natural interpretation of curl-based pushforward at the level of nonlinear differential quantities: cofactor fields represent transported $H(\mathrm{curl})$04-dimensional data, the Jacobian represents transported volume density, and div-curl structure is the mechanism that stabilizes these quantities under weak convergence. In that sense, curl enters not as an explicit transport map but as the exact differential constraint that makes pushforward-like limits mathematically controllable.