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Kernel projection gives dense conforming gauge approximants

Statement

Assume Countable Choice. Let $Ω$ be a bounded connected Lipschitz polyhedron and let ${T_{h}}$ be a shape-regular family of matching affine tetrahedral meshes with maximum cell diameter $h \to 0$ , as in Matching affine simplicial meshes and shape regularity and the mesh hypothesis of Smooth fields are approximated by the lowest-order edge family. For each mesh let $K_{h} = \ker (curl ∣_{V_{h}})$ and $X_{h} = V_{h} \cap K_{h}^{⊥}$ , with the $L^{2}$ inner product. Then for every $u \in X$ and every sequence of mesh sizes $h_{n} \to 0$ , there are $u_{h_{n}} \in X_{h_{n}}$ such that $∥ u_{h_{n}} - u ∥_{H (curl; Ω)} ⟶ 0.$

Facts & Assumptions

Given: Countable Choice, the constant $ϵ > 0$ from the Maxwell definition, $u \in X$ , the finite-dimensional PEC edge spaces $V_{h}$ , and the shape-regular refining mesh family from Smooth fields are approximated by the lowest-order edge family.

[A1]

Countable Choice is inherited from the smooth edge-approximation supplier. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

$H_{0} (curl)$ is the graph-norm closure of $C_{c}^{\infty} (Ω; R^{3})$ . Since $V_{h} \subset V$ and every vector in $K_{h}$ has zero distributional curl, $K_{h} \subset K$ . (Zero tangential trace H(curl) space, PEC Maxwell cavity eigenproblem and full-kernel gauge)

[F2]

For every compactly supported smooth field $w$ , the canonical edge interpolants $I_{h} w$ are in $V_{h}$ for sufficiently small $h$ and converge to $w$ in the $H (curl)$ norm. (Smooth fields are approximated by the lowest-order edge family)

[F3]

Every finite-dimensional subspace of the real $L^{2}$ Hilbert space has an orthonormal basis; a closed subspace has an orthogonal projection. (Every finite-dimensional real or complex inner product space has an orthonormal basis, Orthogonal decomposition by a closed subspace)

[F4]

Since $ϵ$ is a positive constant scalar, the weighted and ordinary $L^{2}$ orthogonal complements of $K$ and $K_{h}$ coincide. (PEC Maxwell cavity eigenproblem and full-kernel gauge)

Proof

Given: Countable Choice, $u \in X$ , and an arbitrary mesh sequence $h_{n} \to 0$ .

1.1A1F1F2givenchoose

By [F1], for every $j \geq 1$ there is $φ_{j} \in C_{c}^{\infty} (Ω; R^{3})$ with $∥ u - φ_{j} ∥_{H (curl)} < 1 / j$ . Countable Choice selects one such field for each $j$ . By [F2], for each fixed $j$ there is an index $N_{j}$ such that for every $n \geq N_{j}$ , $I_{h_{n}} φ_{j} \in V_{h_{n}}$ and $∥ I_{h_{n}} φ_{j} - φ_{j} ∥_{H (curl)} < 1 / j$ ; choose the least such $N_{j}$ , and replace the sequence by increasing thresholds $M_{j} : = \max {j, N_{1}, \dots, N_{j}, M_{j - 1} + 1}$ , with $M_{0} = 0$ .

1.2A1F3givenconstruct

Each $K_{h_{n}}$ is a finite-dimensional subspace of the real $L^{2}$ Hilbert space. By Countable Choice [A1], choose an orthonormal basis for each $K_{h_{n}}$ using [F3]. If a sequence in $K_{h_{n}}$ converges in $L^{2}$ , its coordinates against this finite basis converge, and the limit is the corresponding finite linear combination; thus $K_{h_{n}}$ is closed. By [F3] it has an orthogonal projection $P_{n}$ satisfying $∥ P_{n} y ∥_{2} \leq ∥ y ∥_{2}$ , including $P_{n} = 0$ when $K_{h_{n}} = {0}$ .

2.1step 1.1algebra

For each $n$ with ${j \leq n : M_{j} \leq n} \neq \emptyset$ , let $j (n)$ be its largest member and set $w_{n} : = I_{h_{n}} φ_{j (n)}$ ; for the finitely many initial indices where the set is empty put $w_{n} = 0$ . Since $M_{j} \geq j$ and is strictly increasing, $j (n) \to \infty$ . For all later indices, $∥ w_{n} - u ∥_{H (curl)} < 2 / j (n)$ , so $w_{n} \to u$ in graph norm.

3.1F1F3F4step 2.1step 1.2

Since $K_{h_{n}} \subset K$ by [F1] and $u \in K^{⊥}$ by [F4], $P_{n} u = 0$ . Thus $∥ P_{n} w_{n} ∥_{2} = ∥ P_{n} (w_{n} - u) ∥_{2} \leq ∥ w_{n} - u ∥_{2}$ . Set $u_{h_{n}} : = w_{n} - P_{n} w_{n}$ . Both terms lie in $V_{h_{n}}$ and the projection residual is orthogonal to $K_{h_{n}}$ , so $u_{h_{n}} \in X_{h_{n}}$ .

4.1step 2.1step 3.1algebra∎

Because $curl (P_{n} w_{n}) = 0$ , $∥ u_{h_{n}} - u ∥_{2} \leq 2 ∥ w_{n} - u ∥_{2}$ and $∥ curl (u_{h_{n}} - u) ∥_{2} = ∥ curl (w_{n} - u) ∥_{2}$ . Consequently $∥ u_{h_{n}} - u ∥_{H (curl)} \leq 3 ∥ w_{n} - u ∥_{H (curl)} \to 0$ by step 2.1.

Depends on

Used by

Spectral convergence and no nonzero pollution for PEC edge elements Theorem

Dependency tree · two levels

30 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)