Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Discrete Maxwell energy balance and exact incidence constraints

Physical scope

A conditional finite-dimensional Galerkin deduction for the lowest-order PEC Maxwell semidiscretization defined on a matching tetrahedral mesh with static symmetric uniformly positive material tensors, a symmetric nonnegative conductivity form, continuous forcing, and the stated compatible incidence maps. It gives exact semidiscrete energy and constraint identities, but no continuum convergence claim, no automatic charge-deposition scheme, no claim about actual materials, and no empirical conclusion.

Statement

Assume Countable Choice and the semidiscrete model of PEC Whitney edge-face Maxwell semidiscretization on a nonempty open time interval I. For every t0 in I, initial vectors e(t0)=e0 and b(t0)=b0, and continuous forcing J_ext:I -> L^2(Omega;R3), the finite compatible matrix system for the edge and face coefficients has a unique C1 solution on all of I. Define

$W_{h} (t) : = \frac{1}{2} \int_{Ω} (E_{h} (x, t) \cdot ε (x) E_{h} (x, t) + B_{h} (x, t) \cdot μ (x)^{- 1} B_{h} (x, t)) d x, q (t) : = - G^{T} M_{e} e (t) .$

Then for every t in I,

$\frac{d}{d t} W_{h} (t) = - e (t)^{T} S e (t) - e (t)^{T} j (t), D b (t) = D b (t_{0}), \dot{q} (t) = G^{T} (S e (t) + j (t)) .$

The first identity is the exact semidiscrete field-energy balance in joules; e^T S e and e^T j have units watts, representing discrete conductivity loss and source work. The conserved vector D b has units webers and is the face-to-cell magnetic-divergence coordinate. The coordinate q has units coulombs and is the weak discrete-divergence quantity defined by the finite matrix system; dot q has units amperes. This theorem does not infer a nodal charge-deposition method from it.

Statement references

PEC Whitney edge-face Maxwell semidiscretization

Energy, existence and algebraic constraints

Facts & Assumptions

Given: Countable Choice, the coefficients and mesh of PEC Whitney edge-face Maxwell semidiscretization, an open interval I, t0 in I, initial coefficient vectors e0,b0, and J_ext in C(I;L^2(Omega;R3)).

[F1]

PEC Whitney edge-face Maxwell semidiscretization: the assembled matrices M_e,M_b are symmetric positive definite on nonzero finite-dimensional spaces, S is symmetric positive semidefinite, DC=CG=0, J_ext is continuous into L^2, and the weak Galerkin equations are exactly M_e dot e=C^T M_b b-Se-j and dot b=-Ce. Continuity of each load coordinate j_i follows in step 1.1 from the L^2 pairing.

[F2]

Energy, existence and algebraic constraints: for that finite compatible matrix system and continuous j, every initial state has a unique C1 solution on I, with W'(e,b)=-e^TSe-e^Tj, Db(t)=Db(t0), and dot q=G^T(Se+j) for q=-G^TM_e e.

[A1]

Countable Choice is inherited from the graph-space and real-L^2 conventions in the finite-element definition; the finite-dimensional ODE argument introduces no stronger choice principle. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Proof

1.1givenF1F2A1algebra

By [F1], the semidiscrete Galerkin equations in the chosen edge and face bases are exactly the finite compatible matrix system; the positivity and compatibility hypotheses required by [F2] hold. Countable Choice is the stated graph-space strength [A1]. Each coordinate j_i(t) is continuous because J_ext is continuous in L^2 and phi_i is a fixed L^2 basis function, so Cauchy-Schwarz makes t -> integral(phi_i dot J_ext(t)) continuous. Applying [F2] gives a unique C1 coefficient solution for every initial pair on all of I.

2.1F1F2step 1.1algebra∎

The definitions of M_e and M_b give e^T M_e e=integral(E_h dot epsilon E_h) and b^T M_b b=integral(B_h dot mu^-1 B_h), so W_h is exactly the matrix energy W(e,b) in [F2]. Its derivative is therefore -e^T S e-e^T j. Since S is positive semidefinite, e^T S e>=0; this is a model deduction of discrete dissipation, not an observation about a material. For the same solution established in step 1.1, [F2] also gives D b(t)=D b(t0) and dot q=G^T(S e+j), with q=-G^T M_e e as defined in the Statement. The former is the exact face-to-cell magnetic-divergence constraint; the latter is the stated weak discrete-divergence evolution and does not infer a nodal charge-deposition method.

Discrete Maxwell energy balance and exact incidence constraints

Statement

Statement references

Facts & Assumptions

Proof

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