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Compatible magnetic initial projection for the Maxwell edge method

Statement

Assume the contractible Lipschitz domain and shape-regular mesh-family hypotheses of Uniformly bounded commuting projection for lowest-order Nedelec spaces, with matching affine tetrahedra and the PEC edge space of Edge-face Maxwell finite element semidiscretization. Assume Countable Choice as in that projection interface. Let $B_{0} \in H (div; Ω)$ be piecewise smooth, have matching normal traces, and satisfy $div B_{0} = 0$ . For the approximation bound below, assume all material and field smoothness interfaces are mesh-aligned, with $B_{0}$ and $div B_{0}$ piecewise $C^{1}$ on cell closures and bounded cellwise first derivatives as in Smooth lowest-order Nedelec and RT0 interpolation estimate. Let $π_{h}^{2}$ be the degree-two cochain projection in that interface, let $X_{h} = range C \subset N_{h}^{2}$ , and let $P_{X_{h}}$ be orthogonal projection onto $X_{h}$ for $(u, v)_{μ^{- 1}} = \int_{Ω} u^{T} μ^{- 1} v d x$ . Define $B_{h}^{0} = π_{h}^{2} B_{0} + P_{X_{h}} (B_{0} - π_{h}^{2} B_{0}) .$ Then $B_{h}^{0} \in N_{h}^{2}$ , $D B_{h}^{0} = 0$ , and $(B_{h}^{0} - B_{0}, curl v_{h})_{μ^{- 1}} = 0 (v_{h} \in V_{h}), ∥ B_{h}^{0} - B_{0} ∥_{μ^{- 1}} \leq C h M_{B},$ where $V_{h} = N_{h}^{1} \cap H_{0} (curl; Ω)$ and $M_{B}$ is the maximum of the piecewise first-derivative bounds for $B_{0}$ and $div B_{0}$ used by the smooth interpolation estimate. The error constant is uniform over the shape-regular family when the commuting projection norms are uniformly bounded.

Facts & Assumptions

Given: The finite element spaces, curl matrix $C$ , divergence matrix $D$ , positive magnetic inner product, and the declared Countable Choice branch.

[A1]

Countable Choice is the only choice axiom required by the current bounded commuting projection interface; this proof makes no stronger choice assumption. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The degree-two and degree-three operators $π_{h}^{2}, π_{h}^{3}$ are projections onto $N_{h}^{2}, N_{h}^{3}$ , are uniformly bounded on their de Rham graph spaces, and satisfy $div π_{h}^{2} = π_{h}^{3} div$ . (Uniformly bounded commuting projection for lowest-order Nedelec spaces)

[F2]

The oriented moment matrices represent $curl : V_{h} \to N_{h}^{2}$ and $div : N_{h}^{2} \to N_{h}^{3}$ , with $D C = 0$ . (Edge-face Maxwell finite element semidiscretization, Whitney edge, face, and cell moments commute with grad, curl, and div)

[F3]

Under the stated mesh alignment and cellwise $C^{1}$ bounds, for $B_{0}$ with matching normal traces, the canonical face interpolant $I_{h}^{2} B_{0}$ lies in $N_{h}^{2}$ and has $∥ B_{0} - I_{h}^{2} B_{0} ∥_{H (div)} \leq C h M_{B}$ . (Smooth lowest-order Nedelec and RT0 interpolation estimate)

Proof

1.1A1givenconstructalgebra

The subspace $X_{h}$ is finite dimensional. In a basis of $X_{h}$ , its $μ^{- 1}$ Gram matrix is symmetric positive definite, so the normal equations have one solution for each $B_{0} - π_{h}^{2} B_{0}$ ; this defines $P_{X_{h}}$ . Its value lies in $X_{h} \subset N_{h}^{2}$ , and the definition of orthogonal projection gives $B_{h}^{0} - B_{0} = - (I - P_{X_{h}}) (B_{0} - π_{h}^{2} B_{0}) ⊥_{μ^{- 1}} X_{h}$ . Orthogonal decomposition yields $∥ B_{h}^{0} - B_{0} ∥_{μ^{- 1}} \leq ∥ B_{0} - π_{h}^{2} B_{0} ∥_{μ^{- 1}}$ .

2.1F1F3step 1.1algebra

Since $I_{h}^{2} B_{0} \in N_{h}^{2}$ and $π_{h}^{2}$ is a projection, $π_{h}^{2} I_{h}^{2} B_{0} = I_{h}^{2} B_{0}$ . Thus $B_{0} - π_{h}^{2} B_{0} = (I - π_{h}^{2}) (B_{0} - I_{h}^{2} B_{0})$ . Uniform boundedness of $π_{h}^{2}$ on $H (div)$ , norm equivalence from the coefficient bounds, and [F3] imply $∥ B_{0} - π_{h}^{2} B_{0} ∥_{μ^{- 1}} \leq C h M_{B}$ . Combining with step 1.1 gives the asserted approximation estimate.

3.1F1F2givenstep 1.1algebra∎

The projection identity and [F1] give $D π_{h}^{2} B_{0} = π_{h}^{3} (div B_{0}) = 0$ . Every element of $X_{h}$ is a discrete curl, so [F2] gives $D P_{X_{h}} (B_{0} - π_{h}^{2} B_{0}) = 0$ . Adding the two terms proves $D B_{h}^{0} = 0$ .

Depends on

Used by

Smooth PEC edge finite element Maxwell error from compatible projections Theorem

Dependency tree · two levels

23 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)
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability (standard reference, not scraped)