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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].

Depends on

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Sources

Long Chen, Finite Element Methods for Maxwell Equations, complete lecture notes (standard reference, not scraped)