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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coercive shifted Maxwell Galerkin approximation

Physical scope

A conditional static-coefficient, lossless positive-shift weak Maxwell model on a bounded Lipschitz polyhedron with closure-defined PEC electric space and a shape-regular refining family of conforming lowest-order edge spaces. The theorem gives well-posedness and best-approximation convergence in a dimensionally scaled graph norm. It gives no real-frequency uniqueness, no error rate without further regularity, no assertion that any actual material satisfies the model, and no empirical claim.

Statement

Assume Countable Choice. Let Omega be a nonempty bounded Lipschitz polyhedron in R3 in one SI laboratory frame. Use the real space V:=H_0(curl;Omega), where H_0(curl;Omega) is the graph closure of compactly supported smooth vector fields as defined by the mathematical supplier. Let epsilon,mu:Omega -> Sym_3(R) be static measurable essentially bounded fields with units F m^-1 and H m^-1 and constants 0<a_epsilon<=b_epsilon<infinity and 0<a_mu<=b_mu<infinity such that, almost everywhere,

$a_{ε} ∣ v ∣^{2} \leq v \cdot ε (x) v \leq b_{ε} ∣ v ∣^{2}, a_{μ} ∣ v ∣^{2} \leq v \cdot μ (x) v \leq b_{μ} ∣ v ∣^{2} (v \in R^{3}) .$

Take the lossless branch sigma=0 of the semidiscrete Maxwell model. Fix a reference length ell_0>0 in metres, and equip V with the equivalent, dimensionally homogeneous graph norm

$∥ v ∥_{V, ℓ_{0}}^{2} : = ∥ v ∥_{L^{2} (Ω)}^{2} + ℓ_{0}^{2} ∥ curl v ∥_{L^{2} (Ω)}^{2} .$

Let s>0 be a real frequency-shift parameter in s^-1, let F:V -> R be a bounded real linear functional scaled so that F(v) has units W s^-1 when v has electric-field units V m^-1, and define

$a_{s} (u, v) : = \int_{Ω} (μ^{- 1} curl u \cdot curl v + s^{2} ε u \cdot v) d x .$

Let {T_n}{n>=1} be a shape-regular family of matching affine tetrahedral meshes of closure(Omega), with h_n:=max{K in T_n} diam(K) tending to zero. For each mesh set V_n:=N_{h_n}^1 intersect H_0(curl;Omega), using the PEC Whitney edge space of PEC Whitney edge-face Maxwell semidiscretization. Let

$α : = \min {(b_{μ} ℓ_{0}^{2})^{- 1}, s^{2} a_{ε}}, M : = \max {(a_{μ} ℓ_{0}^{2})^{- 1}, s^{2} b_{ε}} .$

Then there is a unique u in V satisfying a_s(u,v)=F(v) for every v in V. For each n there is a unique Galerkin solution u_n in V_n satisfying a_s(u_n,v_n)=F(v_n) for every v_n in V_n. For every n,

$∥ u - u_{n} ∥_{V, ℓ_{0}} \leq \frac{M}{α} \inf_{v_{n} \in V_{n}} ∥ u - v_{n} ∥_{V, ℓ_{0}},$

and u_n converges to u in the V,ell_0 norm as n tends to infinity. Both alpha and M have units (H m)^-1, so their ratio is dimensionless.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Finite-energy PEC and impedance boundary formulation

PEC Whitney edge-face Maxwell semidiscretization

The real H(curl) graph space on a polyhedron

Lowest-order Whitney and Nedelec de Rham spaces

Zero tangential trace H(curl) space

$L^{2}$ with the integral pairing is a Hilbert space

The Lax--Milgram theorem

Uniformly bounded commuting projection for lowest-order Nedelec spaces

Facts & Assumptions

Given: The coefficient, domain, SI, and closure-defined PEC conventions in the Statement; Countable Choice; and the stated mesh family.

[F1]

Finite-energy PEC and impedance boundary formulation: epsilon and mu are real symmetric static multiplication fields, essentially bounded, with almost-everywhere uniform positive lower and upper bounds. Their SI units are F m^-1 and H m^-1.

[F2]

The real H(curl) graph space on a polyhedron and Zero tangential trace H(curl) space: H(curl;Omega) is the real L^2 graph space of the distributional curl with norm squared ||v||_2^2+||curl v||_2^2; H_0(curl;Omega) is exactly the closure of C_c^\infty(Omega;R3) in that graph norm. The graph closure is not replaced by an unproved trace-kernel identification.

[F3]

The Lax--Milgram theorem: on a real Hilbert space, a bounded coercive real bilinear form and a bounded real linear functional have a unique weak solution.

[F4]

Uniformly bounded commuting projection for lowest-order Nedelec spaces: for this shape-regular refining tetrahedral family there are boundary-corrected projections pi_{h,0}^1:H_0(curl;Omega)->N_h^1 intersect H_0(curl;Omega), identity on each target space and strongly convergent in the graph norm on H_0(curl;Omega). The theorem does not identify these projections with raw edge moments.

[F6]

PEC Whitney edge-face Maxwell semidiscretization and Lowest-order Whitney and Nedelec de Rham spaces: for each mesh in the family the conforming PEC trial space is exactly V_n=N_{h_n}^1 intersect H_0(curl;Omega); the underlying lowest-order space on a finite mesh is finite-dimensional.

[A1]

Countable Choice is the exact choice strength of the imported real L^2 and weak-derivative interfaces; the local Hilbert-space and finite-element arguments add no stronger choice axiom. The Axiom of Countable Choice ( ${A C}_{ω}$ ) $L^{2}$ with the integral pairing is a Hilbert space

Proof

1.1F2A1algebra

The distributional graph of curl is closed: if v_j->v and curl v_j->w in L^2, then for every compactly supported smooth test field phi, integral(v dot curl phi)=lim_j integral(v_j dot curl phi)=lim_j integral(curl v_j dot phi)=integral(w dot phi); hence curl v=w distributionally. L^2 is complete under [A1] and $L^{2}$ with the integral pairing is a Hilbert space, so H(curl;Omega) is Hilbert, and its graph-closed subspace H_0(curl;Omega) is Hilbert. Since ell_0 is fixed and positive, ||.||_{V,ell_0} is equivalent to the defining graph norm and gives the same complete space.

1.2givenF1algebra

From [F1], the eigenvalues of mu^-1 lie between b_mu^-1 and a_mu^-1. Therefore a_s(v,v)>=b_mu^-1||curl v||_2^2+s^2a_epsilon||v||2^2>=alpha||v||{V,ell_0}^2. Cauchy-Schwarz and the upper coefficient bounds give |a_s(u,v)|<=a_mu^-1||curl u||_2||curl v||_2+s^2b_epsilon||u||2||v||2<=M||u||{V,ell_0}||v||{V,ell_0}. Since s has units s^-1, ell_0 has units m, and epsilon and mu have units F m^-1 and H m^-1 by [F1], both constants have units (H m)^-1; alpha>0 and M<infinity by the stated strict bounds.

2.1F3F6step 1.1step 1.2given

Apply [F3] to the real Hilbert space and form of steps 1.1-1.2 to obtain the unique u. By [F6], each V_n is finite-dimensional and closed in V; restricting the same bounded coercive form and F to V_n gives the unique Galerkin solution u_n, including the zero-space case. If F=0, the zero vector is the solution on V and on every V_n by uniqueness.

3.1step 1.2step 2.1algebra

Put e_n:=u-u_n. For any w_n in V_n, Galerkin orthogonality gives a_s(e_n,u_n-w_n)=0, so a_s(e_n,e_n)=a_s(e_n,u-w_n). Coercivity and boundedness from step 1.2 imply alpha||e_n||{V,ell_0}^2<=a_s(e_n,e_n)<=M||e_n||{V,ell_0}||u-w_n||_{V,ell_0}. If e_n=0 the asserted estimate is immediate; otherwise divide by alpha||e_n|| and take the infimum over w_n.

4.1F4step 3.1algebra∎

By [F4], pi_{h_n,0}^1 u belongs to V_n and converges to u in the unscaled H(curl) graph norm; equivalence of the fixed ell_0-scaled norm gives ||u-pi_{h_n,0}^1u||_{V,ell_0}->0. Thus the best-approximation infimum in step 3.1 tends to zero, and the error estimate proves u_n->u in V,ell_0. No interpolation by pointwise edge restrictions is used.

Coercive shifted Maxwell Galerkin approximation

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources