ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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A CFL-violating finite-element iterate is rejected before inversion

Example

Let $Ω = (0, 1)^{2}$ and triangulate its square by joining the center $O = (1 / 2, 1 / 2)$ to the four corners. Use continuous piecewise-affine finite elements with zero boundary values and row-sum lumped mass. The only free nodal basis function is $ϕ$ , with value $1$ at $O$ and $0$ at each corner. The one-coordinate semidiscrete heat equation is $m_{h} \dot{U} + k_{h} U = 0$ , where $m_{h} = 1 / 3$ and $k_{h} = 4$ . Treat the initial coefficient $q = U^{0}$ as the inversion parameter, take $Θ_{h} = R$ and $Q_{h} = {1}$ , and let the one-step state $u = U^{1}$ satisfy $R_{h} (q, u) = u - (1 - 12 Δ t) q = 0$ with observation $G_{h} (q, u) = u$ . A forward-Euler step of size $Δ t = 1 / 4$ sends $U^{0}$ to $U^{1} = (1 - 12 Δ t) U^{0} = - 2 U^{0}$ . Its scalar stability condition is $∣ 1 - 12 Δ t ∣ \leq 1$ , equivalently $0 \leq Δ t \leq 1 / 6$ . Use the exact projection $P_{Q_{h}} (z) = 1$ , exact state/adjoint and objective-interval routines, $y^{δ} = 0$ , $q_{0} = 1$ , $α = 1$ , $δ = 0$ , $κ = 1$ , $τ_{g} = 1$ , $L = 1$ , $τ_{u} = τ_{p} = 0$ , and set $N_{i t e r} = 1$ , $N_{s o l v e} = 2$ , $N_{e v a l} = 1$ , and $N_{b a c k t r a c k} = 1$ . The declared admissibility predicate checks this explicit-step stability condition before solving; at $q_{i n i t} = 1$ it rejects the time step, so the inversion algorithm returns inadmissible before a state solve.

The diagram shows the four exact triangular cells and the only interior degree of freedom. It is illustrative; the mass and stiffness calculations in the verification use the stated vertices and affine basis.

Example references

Matching affine simplicial mesh convention

Finite-element spaces and degrees of freedom

Declared PDE and CFL admissibility contract

Finite-budget inversion algorithm

Sound inadmissible status

Facts & Assumptions

Given: The four-triangle mesh, zero boundary degrees of freedom, the center basis function $ϕ$ , row-sum lumped mass, the scalar semidiscrete heat equation, parameter set $Q_{h} = {1}$ with $q = U^{0}$ , state residual $R_{h} (q, u) = u - (1 - 12 Δ t) q$ , observation $G_{h} (q, u) = u$ , the stated exact routines and finite algorithm inputs, and $Δ t = 1 / 4$ .

[F1]

A matching affine simplicial mesh is a finite face-to-face cover with disjoint simplex interiors (Matching affine simplicial meshes and shape regularity).

[F2]

A unisolvent finite element gives a unique shape function for each nodal degree of freedom (Reference finite elements and unisolvent degrees of freedom).

[F3]

The algorithm tests the declared admissibility predicate before any PDE solve and returns inadmissible when it is false (Finite-budget finite-element PDE Tikhonov inversion, Finite-element PDE solver admissibility and certificate contract).

[F4]

The theorem verifies that the algorithm's inadmissible status is returned only on a failed admissibility check (Finite-element PDE Tikhonov statuses are sound and terminate).

Verification

Given: The data in the Example.

1.1givenF1F2algebra

Each of the four triangles has area $1 / 4$ . On each triangle, $ϕ$ is the affine barycentric coordinate of the center, so its gradient has magnitude $2$ . The row-sum lumped mass contribution at the center is $∣ K ∣ / 3 = 1 / 12$ on each triangle, hence $m_{h} = 4 / 12 = 1 / 3$ . The stiffness is $k_{h} = \sum_{K} \int_{K} ∣ \nabla ϕ ∣^{2} = 4 \cdot 4 \cdot (1 / 4) = 4$ .

2.1givenstep 1.1algebra

Dividing the semidiscrete equation by $m_{h}$ gives $\dot{U} = - (k_{h} / m_{h}) U = - 12 U$ . By the specified forward-Euler update, $U^{1} = U^{0} + Δ t {\dot{U}}^{0} = (1 - 12 Δ t) U^{0} = - 2 U^{0}$ . The scalar amplification has modulus at most one exactly when $- 1 \leq 1 - 12 Δ t \leq 1$ , which is equivalent to $0 \leq Δ t \leq 1 / 6$ ; $Δ t = 1 / 4$ violates it.

3.1givenF3step 2.1∎

The model's admissibility predicate requires this explicit-step condition. Since the given step fails it, [F3] shows that the algorithm returns inadmissible before calling either state or adjoint solver. This conclusion concerns the declared discrete stability condition, not a stability theorem for the continuous heat equation.

Depends on

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