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Semidiscrete edge-face Maxwell energy conservation

Statement

Assume Countable Choice only as inherited from the real $H (div)$ /Piola finite-element space interfaces in Edge-face Maxwell finite element semidiscretization. For the lossless source-free semidiscretization $M_{e} \dot{e} - C^{T} M_{b} b = 0, \dot{b} + C e = 0,$ the weighted energy $H_{h} (t) = \frac{1}{2} ∥ e (t) ∥_{M_{e}}^{2} + \frac{1}{2} ∥ b (t) ∥_{M_{b}}^{2}$ is constant in time. The local matrix calculation uses no further choice principle.

Facts & Assumptions

Given: The finite-dimensional semidiscrete fields, symmetric positive definite matrices, and the source-free equations above.

[A1]

The Countable Choice Axiom is the principle that every countable family of nonempty sets has a choice function; it is included only because the imported FE-space and Piola definitions declare it. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

$M_{e}, M_{b}$ are symmetric positive definite and the semidiscrete equations are $M_{e} \dot{e} = C^{T} M_{b} b$ and $M_{b} \dot{b} = - M_{b} C e$ . (Edge-face Maxwell finite element semidiscretization)

Proof

1.1A1F1algebra

On a singleton time interval the energy is constant immediately; assume henceforth that the time interval is nondegenerate. In the declared Countable Choice branch [A1], the FE bases and matrices in [F1] are fixed finite objects; the calculation below makes no selection. Differentiating the electric part gives $\frac{d}{d t} \frac{1}{2} e^{T} M_{e} e = e^{T} M_{e} \dot{e} = e^{T} C^{T} M_{b} b$ .

2.1F1step 1.1algebra∎

Differentiating the magnetic part gives $\frac{d}{d t} \frac{1}{2} b^{T} M_{b} b = b^{T} M_{b} \dot{b} = - b^{T} M_{b} C e = - e^{T} C^{T} M_{b} b$ , because the scalar transpose equals itself. The two derivatives cancel, so $H_{h}^{'} (t) = 0$ throughout the interval of existence.

Depends on

Used by

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Sources

Long Chen, Finite Element Methods for Maxwell Equations, complete lecture notes (standard reference, not scraped)
John B. Schneider, Understanding the Finite-Difference Time-Domain Method (standard reference, not scraped)