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Discrete de Rham complexes and edge-element foundations — Examples

1 · Prerequisites

2 · Summary

The examples compute one oriented Nedelec edge circulation and its face curl flux, exhibit a zero-curl nodal vector mode outside the compatible nodal-gradient image, and give an azimuthal curl-free field with nonzero circulation on a solid torus. The nodal example diagnoses a mismatch in that chosen finite element pair; the handle example shows why a cohomology or harmonic component must be retained on noncontractible domains.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

An oriented Nedelec edge circulation on one tetrahedron

Example

On the reference tetrahedron with vertices $x_{0} = 0$ , $x_{1} = e_{1}$ , $x_{2} = e_{2}$ , $x_{3} = e_{3}$ , take the Whitney edge field $ϕ_{01} = λ_{0} \nabla λ_{1} - λ_{1} \nabla λ_{0}$ and orient the face $[0, 1, 2]$ by the vertex order shown. Compute the circulation on $[0, 1]$ and the curl flux through $[0, 1, 2]$ .

Facts & Assumptions

Given: The local edge basis and oriented edge and face moments of Lowest-order Whitney and Nedelec de Rham spaces, their unisolvence from Lowest-order Nedelec edge circulation unisolvence, and Stokes commutation from Whitney edge, face, and cell moments commute with grad, curl, and div.

[F1]

The barycentric edge field is $ϕ_{01} = λ_{0} \nabla λ_{1} - λ_{1} \nabla λ_{0}$ , and its edge circulation is the canonical degree of freedom. (Lowest-order Whitney and Nedelec de Rham spaces)

[F2]

The six local edge circulations form the dual basis for the Whitney edge fields. (Lowest-order Nedelec edge circulation unisolvence)

[F3]

On the oriented face $[0, 1, 2]$ , the curl flux equals the signed sum of its three boundary edge circulations. (Whitney edge, face, and cell moments commute with grad, curl, and div)

Verification

1.1F1constructalgebra

In the reference coordinates, $λ_{1} = x_{1}$ , $λ_{0} = 1 - x_{1} - x_{2} - x_{3}$ , so $ϕ_{01} = (1 - x_{2} - x_{3}, x_{1}, x_{1})$ . Along $[0, 1]$ , parameterized by $x (t) = (t, 0, 0)$ , this field has tangent component $1$ ; therefore its oriented circulation is $\int_{0}^{1} 1 d t = 1$ .

2.1F1F2F3step 1.1algebra∎

Differentiating the displayed polynomial gives $curl ϕ_{01} = (0, - 2, 2)$ . The oriented face $[0, 1, 2]$ lies in $x_{3} = 0$ with normal $e_{3}$ and area $1 / 2$ , so its curl flux is $2 \cdot (1 / 2) = 1$ . Its oriented boundary is $[1, 2] - [0, 2] + [0, 1]$ ; by [F2], the first two circulations vanish and the last is one, agreeing with the flux by [F3].

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A nodal curl-free mode outside the compatible gradient image

Example

On one tetrahedron, compare the naive vector nodal space $[P_{1} (K)]^{3}$ with the compatible scalar-to-edge gradient map $\nabla W_{h}^{0}$ of Lowest-order Whitney and Nedelec de Rham spaces. The vector field $v (x) = x_{1} e_{1}$ lies in the nodal vector space, has zero curl, and is not the gradient of a continuous piecewise-affine scalar. Thus the naive nodal curl-curl matrix has a zero-energy mode outside that compatible gradient image. This identifies a discrete kernel mismatch; it does not claim that gradients are absent from the continuum curl kernel.

Facts & Assumptions

Given: Countable Choice for the real $L^{2}$ gauge convention; a reference tetrahedron, the scalar nodal space and compatible edge space of Lowest-order Whitney and Nedelec de Rham spaces, and the gauge formulation of Kernel quotient and gauge for discrete curl-curl.

[A1]

Countable Choice is assumed by the real $L^{2}$ product and the stable gauge supplier; no full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The scalar nodal space consists of affine functions on the tetrahedron, so every member has constant gradient. (Lowest-order Whitney and Nedelec de Rham spaces)

[F2]

The pure curl-curl energy vanishes on its curl kernel, and the mixed gauge formulation fixes a representative by orthogonality to that kernel. (Kernel quotient and gauge for discrete curl-curl, Stable mixed curl-curl formulation with kernel gauge)

Verification

1.1A1givenF2algebra

The field $v (x) = (x_{1}, 0, 0)$ is vector-valued affine, so it belongs to $[P_{1} (K)]^{3}$ . Its componentwise curl is zero because its only nonzero component depends only on $x_{1}$ . Hence $(curl v, curl v)_{L^{2} (K)} = 0$ .

2.1A1F1F2step 1.1algebra∎

By [F1], every $q_{h} \in W_{h}^{0} = P_{1} (K)$ has constant gradient, while $v$ is nonconstant on the interior of $K$ . Thus $v \notin \nabla W_{h}^{0}$ . The kernel of the naive nodal curl-curl energy is therefore strictly larger than the compatible discrete gradient image, and its representative is not unique without a gauge as in [F2].

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A handle-domain harmonic field with nonzero circulation

Example

Let $Ω = {(x, y, z) : (\sqrt{x^{2} + y^{2}} - R)^{2} + z^{2} < r^{2}}$ be a solid torus with $R > r > 0$ . Define $u (x, y, z) = (- \frac{y}{x^{2} + y^{2}}, \frac{x}{x^{2} + y^{2}}, 0) .$ This is a smooth curl-free and divergence-free field on $\overline{Ω}$ , tangent to the boundary, whose circulation around the core circle is $2 π$ . Here “harmonic” means curl and divergence both vanish with zero normal boundary component. It is a concrete nonzero-period field caused by the handle; the nonzero period proves it is not the gradient of a single-valued $C^{1}$ potential. It is the continuous model for the nontrivial cohomology components described in Discrete de Rham cohomology and contractible-mesh exactness.

Facts & Assumptions

Given: Countable Choice for the graph-space and real $L^{2}$ conventions; $R > r > 0$ , the H(curl) convention of The real H(curl) graph space on a polyhedron, the topology-dependent cohomology convention of Discrete de Rham cohomology and contractible-mesh exactness, and the core loop $γ (θ) = (R \cos θ, R \sin θ, 0)$ , $0 \leq θ \leq 2 π$ .

[A1]

Countable Choice is assumed by the cited H(curl) and cohomology interfaces; the explicit field and circulation computation use no stronger choice axiom. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

H(curl) consists of square-integrable vector fields with square-integrable distributional curl. (The real H(curl) graph space on a polyhedron)

[F2]

The classical coordinate divergence and curl are given by the usual partial-derivative formulas for a $C^{1}$ vector field. (Divergence and curl of a $C^{1}$ vector field)

[F3]

On a noncontractible realization, curl-free edge fields modulo gradients have the first cohomology dimension and must be retained as a harmonic component. (Discrete de Rham cohomology and contractible-mesh exactness)

[F4]

For a smooth scalar on a smooth segment, the integral of its derivative equals the endpoint difference. (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz)

Verification

1.1A1F1F2givenalgebra

On $\overline{Ω}$ , $\sqrt{x^{2} + y^{2}} \geq R - r > 0$ , so every component of $u$ and its first derivatives are bounded; hence $u \in L^{2} (Ω; R^{3})$ and its classical curl is also in $L^{2}$ . Direct differentiation by [F2] gives $div u = 2 x y / (x^{2} + y^{2})^{2} - 2 x y / (x^{2} + y^{2})^{2} = 0$ and $curl u = 0$ . Thus [F1] places $u$ in H(curl) with zero curl.

2.1A1F2F3F4givenalgebra∎

Write $e_{θ} = (- \sin θ, \cos θ, 0)$ . On the torus boundary the normal is a linear combination of the meridional radial vector $(\cos θ, \sin θ, 0)$ and $e_{3}$ , whereas $u = e_{θ} / \sqrt{x^{2} + y^{2}}$ ; hence $u \cdot n = 0$ . Along the core curve, $u (γ (θ)) = e_{θ} / R$ and $γ^{'} (θ) = R e_{θ}$ , so $\int_{γ} u \cdot d ℓ = \int_{0}^{2 π} 1 d θ = 2 π$ . A single-valued $C^{1}$ potential would have zero integral around every closed curve by [F4], so this field is not such a gradient and represents a nonzero period. This is the type of harmonic component retained by [F3].

Discrete de Rham complexes and edge-element foundations — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

An oriented Nedelec edge circulation on one tetrahedron

Example

Facts & Assumptions

Verification

A nodal curl-free mode outside the compatible gradient image

Example

Facts & Assumptions

Verification

A handle-domain harmonic field with nonzero circulation

Example

Facts & Assumptions

Verification

Sources