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The PEC full-kernel gauge is infinite-dimensional

Statement

Let $Ω \subset R^{3}$ be any nonempty open set, let $V = H_{0} (curl; Ω)$ , let $K = {u \in V : curl u = 0}$ , and put $X = V \cap K^{⊥}$ using the $L^{2}$ inner product. Then $X$ is infinite-dimensional.

Facts & Assumptions

Given: a nonempty open $Ω \subset R^{3}$ , the distributional curl convention, and a ball $B_{r} (x_{0}) \subset Ω$ .

[F1]

A nonempty open subset of Euclidean space contains a nondegenerate ball.

[F2]

For every pair $0 < ρ < R$ there is a smooth bump equal to one on the smaller closed ball and supported in the larger ball. (A smooth bump between concentric Euclidean balls)

[F3]

For $k \in H (curl)$ , $\int_{Ω} k \cdot curl A d x = \int_{Ω} curl k \cdot A d x$ for every $A \in C_{c}^{\infty} (Ω; R^{3})$ , by the distributional definition. (The real H(curl) graph space on a polyhedron)

Proof

Given: the nonempty open set, a ball $B_{r} (x_{0}) \subset Ω$ , and the full-kernel gauge $X$ .

1.1F1F2givenchooseconstruct

For $j \geq 1$ , set $c_{j} = x_{0} + r (1 - 2^{- j}) e_{1}$ and $r_{j} = r 2^{- j - 4}$ . The balls $B_{j} = B_{r_{j}} (c_{j})$ lie in $B_{r} (x_{0})$ : $∣ c_{j} - x_{0} ∣ + r_{j} = r (1 - 15 \cdot 2^{- j - 4}) < r$ . If $l > j$ , $∣ c_{l} - c_{j} ∣ = r (2^{- j} - 2^{- l}) \geq r 2^{- j - 1}$ , while $r_{j} + r_{l} \leq 3 r 2^{- j - 5} < r 2^{- j - 1}$ , so these balls are pairwise disjoint. By [F2], choose $ρ \in C_{c}^{\infty} (B_{1} (0))$ equal to one near the origin. Set $ψ (ξ) = ξ_{1} ρ (ξ)$ , so $\partial_{1} ψ (0) = 1$ , and put $ψ_{j} (x) = ψ ((x - c_{j}) / r_{j})$ and $A_{j} = (0, 0, ψ_{j})$ . Then $v_{j} : = curl A_{j} = (\partial_{2} ψ_{j}, - \partial_{1} ψ_{j}, 0)$ is a nonzero smooth vector field supported in $B_{j}$ .

2.1F3step 1.1algebra∎

Each $v_{j}$ has compact support in $Ω$ , hence $v_{j} \in V$ . For every $k \in K$ , [F3] gives $(v_{j}, k)_{L^{2}} = \int_{Ω} k \cdot curl A_{j} d x = \int_{Ω} curl k \cdot A_{j} d x = 0$ . Thus $v_{j} \in X$ . Pairwise disjoint supports and $v_{j} \neq 0$ imply linear independence: restricting any finite vanishing linear combination to $B_{j}$ forces its $j$ th coefficient to be zero. The gauge space therefore contains infinitely many linearly independent vectors and is infinite-dimensional.

Depends on

Used by

Spectral convergence and no nonzero pollution for PEC edge elements Theorem

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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