LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

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.

The midpoint-conductivity Yee electric solve is well defined

Statement

Let $M_{e}$ be the positive definite electric mass matrix, let $S$ be the positive semidefinite conductivity matrix, and let $Δ t > 0$ . Then $M_{e} + (Δ t / 2) S$ is symmetric positive definite, hence invertible. Thus each midpoint-conductivity Yee electric row has a unique solution.

Facts & Assumptions

Given: The material matrices and time-step convention of Oriented Maxwell cochains, masses, and the Yee grid.

[F1]

$M_{e}$ is symmetric positive definite and $S$ is symmetric positive semidefinite. (Oriented Maxwell cochains, masses, and the Yee grid)

Proof

1.1F1givenalgebra

The matrix $M_{e} + (Δ t / 2) S$ is symmetric because both summands are symmetric. For each nonzero $x$ , its quadratic form is $x^{T} M_{e} x + (Δ t / 2) x^{T} S x > 0 + 0 = 0$ , using $Δ t > 0$ and the two definiteness hypotheses. Hence it is positive definite.

2.1step 1.1algebra∎

A positive definite square matrix has trivial kernel: if $(M_{e} + (Δ t / 2) S) x = 0$ , then its quadratic form at $x$ is zero and step 1.1 forces $x = 0$ . A linear map from a finite-dimensional space to itself with trivial kernel is bijective, so the solve exists uniquely for every right-hand side.

Depends on

Oriented Maxwell cochains, masses, and the Yee grid

Used by

Lossless or midpoint-conductive Yee leapfrog update Algorithm

Dependency tree · two levels

2 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

John B. Schneider, Understanding the Finite-Difference Time-Domain Method (standard reference, not scraped)