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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].

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Sources

Long Chen, Finite Element Methods for Maxwell Equations, complete lecture notes (standard reference, not scraped)
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability (standard reference, not scraped)