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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].

Depends on

Used by

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Dependency tree · two levels

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Sources

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