LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Unique solvability of an implicit θ heat step

Statement

For every finite two-dimensional rectangular Dirichlet grid, $Δ t > 0$ , $θ \in [0, 1]$ , and interior right-hand side (including the known source and boundary contributions), the θ step of Fully discrete θ heat finite-difference scheme has a unique interior solution. Its implicit matrix is $I + θ Δ t A_{h}$ , where $A_{h}$ is the zero-boundary operator.

Facts & Assumptions

Given: A finite rectangular interior grid, $Δ t > 0$ , $θ \in [0, 1]$ , and the θ update of Fully discrete θ heat finite-difference scheme.

[F1]

Fully discrete θ heat finite-difference scheme writes the Dirichlet step with unknown terms $(I + θ Δ t A_{h}) U^{n + 1}$ ; all source and prescribed boundary rows are known right-hand-side terms.

[F2]

Symmetry and positivity of the rectangular Dirichlet heat operator: $⟨ A_{h} v, v ⟩_{h} \geq 0$ for every zero-boundary interior vector $v$ .

Proof

1.1F1given

If $θ = 0$ , the implicit matrix in [F1] is the identity, so for every right-hand side there is exactly one solution, namely that right-hand side.

1.2F1F2givenalgebra

Suppose $θ > 0$ and $(I + θ Δ t A_{h}) v = 0$ . Taking the weighted inner product with $v$ gives $∥ v ∥_{h}^{2} + θ Δ t ⟨ A_{h} v, v ⟩_{h} = 0$ . By [F2] the second term is nonnegative, so $∥ v ∥_{h}^{2} = 0$ and $v = 0$ .

2.1step 1.2algebra∎

The interior vector space is finite dimensional and the matrix is square. Its zero kernel from step 1.2 means its rank equals the dimension by rank-nullity; hence its range is the whole space. Thus every right-hand side has exactly one solution.

Remarks

This is finite-dimensional solvability only; it makes no condition-number or mesh-uniform iterative-solver claim. No choice axiom is used.

Depends on

Used by

Dependency tree · two levels

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Sources

Endre Süli, B6.1 Numerical Solution of Partial Differential Equations: A brief introduction to the theory of finite difference approximation of PDEs (standard reference, not scraped)
Benjamin Seibold, MIT 18.336, Lecture 14: Von Neumann Stability Analysis (Spring 2009) (standard reference, not scraped)
Martin Krosche, Hermann G. Matthies, and Oliver Kayser-Herold, Lecture Notes: Introduction to PDEs and Numerical Methods, Technical University Braunschweig, Revision 1.1 (2008), hosted by University of Jyväskylä (standard reference, not scraped)