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Finite-volume diffusion on admissible meshes — Examples

1 · Prerequisites

Finite-volume diffusion on admissible meshes

2 · Summary

These examples calculate an exact shared flux on a two-cell rectangular Voronoï mesh, exhibit a nonorthogonal face where the naive two-point formula produces a spurious transverse flux, and check the affine-exact mimetic reconstruction on a skew parallelogram with an anisotropic tensor. A final one-dimensional interface calculation compares harmonic and arithmetic coefficient averages and verifies the exact resistance-weighted flux across a discontinuous positive coefficient.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Harmonic TPFA flux on a rectangular Voronoï mesh

Example

Take $Ω = (0, 2) \times (0, 1)$ and the two rectangular cells $K = (0, 1) \times (0, 1)$ and $L = (1, 2) \times (0, 1)$ . Their centers are $(1 / 2, 1 / 2)$ and $(3 / 2, 1 / 2)$ , so the shared vertical face is orthogonal to the center segment. Set $λ_{K} = λ_{L} = 1$ , use $u (x, y) = x$ , and impose its exact face-center values on the six cell faces on the outer boundary. The exact source is $f = 0$ .

Facts & Assumptions

Given: The two-cell mesh and data above, and the TPFA transmissibilities and signed fluxes from Orthogonal TPFA diffusion with harmonic face transmissibility.

[F1]

Finite-volume cell-face meshes and geometric regularity: $∣ K ∣ = ∣ L ∣ = 1$ ; the shared face has length $1$ , and each outer face has length $1$ .

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: $T_{F} = ∣ F ∣ / (d_{K, F} / λ_{K} + d_{L, F} / λ_{L})$ on an interior face, and $T_{K, F} = ∣ F ∣ λ_{K} / d_{K, F}$ on a boundary face.

Verification

1.1F1F2givenalgebra

The center-to-center distance across the shared face is $1$ , with half-distances $d_{K, F} = d_{L, F} = 1 / 2$ . Thus $T_{F} = 1$ . The left and right boundary half-distances are $1 / 2$ , so their transmissibilities are $2$ ; top and bottom center-to-face displacements are vertical and hence orthogonal to those faces. The mesh satisfies the TPFA orthogonality requirements.

2.1F1F2step 1.1algebra∎

The cell values are $u_{K} = 1 / 2$ and $u_{L} = 3 / 2$ . At the shared face, $Q_{K, F} = 1 (1 / 2 - 3 / 2) = - 1$ and $Q_{L, F} = 1$ . On the left boundary the outward flux is $2 (1 / 2 - 0) = 1$ , and on the right boundary it is $2 (3 / 2 - 2) = - 1$ . The top and bottom boundary fluxes vanish because their face-center values equal the adjacent cell-center values. Each cell balance therefore sums to zero, and the shared flux matches the exact outward diffusive flux $- \nabla u \cdot (1, 0) = - 1$ from $K$ .

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A nonorthogonal two-point flux gives a false transverse flux

Statement refuted

The Euclidean two-point flux formula $Q_{K, F} = ∣ F ∣ (u_{K} - u_{L}) / ∣ x_{L} - x_{K} ∣$ is consistent for isotropic diffusion on every cell-face mesh, including nonorthogonal meshes.

Witness

Let $K$ be the triangle with vertices $(- 1, 0), (0, 0), (0, 1)$ and $L$ the triangle with vertices $(0, 0), (1, 1), (0, 1)$ . They share the vertical face $F$ from $(0, 0)$ to $(0, 1)$ . The union is a parallelogram and the two cell centroids are $x_{K} = (- 1 / 3, 1 / 3)$ and $x_{L} = (1 / 3, 2 / 3)$ , so their difference is not normal to $F$ . Set $λ = 1$ and evaluate the naive algebraic extension of the TPFA formula on $u (x, y) = y$ .

Facts & Assumptions

Given: The two triangles, their common face, scalar diffusion coefficient $1$ , the affine function $u (x, y) = y$ , and the Euclidean-distance two-point expression in the refuted claim.

[F1]

Finite-volume cell-face meshes and geometric regularity: the common face has length $1$ , outward normal from $K$ equal to $(1, 0)$ , and the cell centers are the centroids of the triangles.

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: when applied under its orthogonality hypothesis, the TPFA formula uses transmissibility $∣ F ∣ / ∣ x_{L} - x_{K} ∣$ for $λ = 1$ . Here we deliberately evaluate that same algebraic expression outside its admissible-mesh hypothesis.

Verification

1.1F1givenalgebra

The triangle-centroid formula gives $x_{K} = (- 1 / 3, 1 / 3)$ and $x_{L} = (1 / 3, 2 / 3)$ , hence $x_{L} - x_{K} = (2 / 3, 1 / 3)$ and $∣ x_{L} - x_{K} ∣ = \sqrt{5} / 3$ . The center line is not perpendicular to the vertical face, whose normal is $(1, 0)$ .

1.2F1givenalgebra

The exact normal derivative of $u (x, y) = y$ on $F$ is $\nabla u \cdot (1, 0) = 0$ , so the exact outward diffusive flux from $K$ is zero. The cell-center values are $u_{K} = 1 / 3$ and $u_{L} = 2 / 3$ .

2.1F2step 1.1step 1.2algebra∎

The nonorthogonal two-point expression gives $Q_{K, F} = \frac{1}{\sqrt{5} / 3} (\frac{1}{3} - \frac{2}{3}) = - \frac{1}{\sqrt{5}} \neq 0.$ It disagrees with the exact zero normal flux in step 1.2. Thus this formula is inconsistent on the stated nonorthogonal face, refuting the claim.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

Harmonic transmissibility across a one-dimensional material interface

Example

Let $Ω = (0, 1)$ with cells $K = (0, 1 / 2)$ and $L = (1 / 2, 1)$ , chosen centers $x_{K} = 1 / 4$ and $x_{L} = 3 / 4$ , cell coefficients $λ_{K} = 1$ and $λ_{L} = 9$ , and boundary values $u (0) = 0$ , $u (1) = 1$ . The stationary continuous solution has constant flux across the material interface. Compare the TPFA transmissibility using the two half-cell resistances with the transmissibility obtained by substituting the arithmetic coefficient mean.

Facts & Assumptions

Given: The two interval cells, piecewise-constant coefficient, Dirichlet boundary data, and the face transmissibility formula from Orthogonal TPFA diffusion with harmonic face transmissibility.

[F1]

The centers $x_{K} = 1 / 4$ and $x_{L} = 3 / 4$ are the choices specified in this example, permitted by Finite-volume cell-face meshes and geometric regularity; the shared interface is $x = 1 / 2$ .

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: the interface transmissibility is $T_{F} = ∣ F ∣ / (d_{K, F} / λ_{K} + d_{L, F} / λ_{L})$ , with one shared flux $T_{F} (u_{K} - u_{L})$ outward from $K$ .

Verification

1.1givenF1algebra

The exact solution is piecewise affine. The constant $c$ in $λ u^{'} = c$ satisfies $1 = c (\frac{1}{2} / 1 + \frac{1}{2} / 9)$ , hence $c = 9 / 5$ . Thus $u (1 / 2) = 9 / 10$ , $u_{K} = u (1 / 4) = 9 / 20$ , and $u_{L} = u (3 / 4) = 19 / 20$ . The outward diffusive flux from $K$ is $- λ_{K} u_{K}^{'} = - 9 / 5$ and equals the outward flux from $L$ with the opposite orientation.

2.1F1F2step 1.1algebra

The half-distances are both $1 / 4$ , so [F2] gives $T_{F} = 1 / (1 / 4 + 1 / 36) = 18 / 5$ . The TPFA flux is $(18 / 5) (9 / 20 - 19 / 20) = - 9 / 5$ , exactly the continuous interface flux.

3.1F1step 1.1algebra∎

The arithmetic mean coefficient is $(1 + 9) / 2 = 5$ and the center distance is $1 / 2$ , which would give transmissibility $5 / (1 / 2) = 10$ . Its flux is $10 (9 / 20 - 19 / 20) = - 5$ , not $- 9 / 5$ . This finite interface calculation verifies that the resistance-weighted harmonic transmissibility is essential for this discontinuous-coefficient example.

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Affine anisotropic flux on a skew parallelogram

Example

Take the single parallelogram cell with vertices $(0, 0), (2, 0), (3, 1), (1, 1)$ , tensor $A = diag (2, 1)$ , and affine state $u (x, y) = x + 2 y$ . Prescribe on each boundary face its exact face mean. The cell center is $(3 / 2, 1 / 2)$ and the face centers, in bottom/right/top/left order, are $(1, 0), (5 / 2, 1 / 2), (2, 1), (1 / 2, 1 / 2)$ . This cell is skew: its right and left face normals are not parallel to the vectors from the cell center to those face centers.

Facts & Assumptions

Given: The skew parallelogram, $A = diag (2, 1)$ , the affine function $u = x + 2 y$ , and the hybrid mimetic reconstruction and flux conventions.

[F1]

Hybrid mimetic finite-volume diffusion on general meshes: $G_{K} u = ∣ K ∣^{- 1} \sum_{F} ∣ F ∣ u_{F} n_{K, F}$ , the residual is $u_{F} - u_{K} - G_{K} u \cdot (x_{F} - x_{K})$ , and the outward mimetic flux is $- A_{K} (u, ϕ_{K, F})$ .

[F2]

Coercivity, monotonicity, conservation, and affine exactness of the mimetic schemes: for affine states the mimetic flux is exactly $- ∣ F ∣ (A \nabla u) \cdot n_{K, F}$ .

Verification

1.1F1givenalgebra

The cell area is $∣ K ∣ = 2$ . The face data $(∣ F ∣, x_{F}, n_{K, F})$ , in bottom/right/top/left order, are $(2, (1, 0), (0, - 1)), (\sqrt{2}, (5 / 2, 1 / 2), (1, - 1) / \sqrt{2}), (2, (2, 1), (0, 1)), (\sqrt{2}, (1 / 2, 1 / 2), (- 1, 1) / \sqrt{2}) .$ Their weighted normals sum to $(0, 0)$ and $\sum_{F} ∣ F ∣ n_{K, F} \otimes x_{F} = 2 I_{2}$ , so the cell satisfies the face-moment closure in [F1].

2.1F1step 1.1algebra

The cell and face means of $u$ are $u_{K} = 5 / 2$ and $(u_{F}) = (1, 7 / 2, 4, 3 / 2)$ . Substitution into [F1] gives $\sum_{F} ∣ F ∣ u_{F} n_{K, F} = (2, 4)$ and hence $G_{K} u = (1, 2) = \nabla u$ . Each residual is zero: for example, on the bottom face $u_{F} - u_{K} = - 3 / 2 = G_{K} u \cdot ((- 1 / 2, - 1 / 2))$ , and the same dot-product identity holds for the other three face centers.

3.1F2givenstep 2.1algebra∎

We have $A \nabla u = (2, 2)$ . By [F2], the outward integrated fluxes in bottom/right/top/left order are $4, 0, - 4, 0$ , respectively. Their sum is zero, which is the exact cell balance for the affine state since its flux is constant and has zero divergence. This verifies affine consistency and local conservation on this skew, anisotropic cell.

Finite-volume diffusion on admissible meshes — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Harmonic TPFA flux on a rectangular Voronoï mesh

Example

Facts & Assumptions

Verification

A nonorthogonal two-point flux gives a false transverse flux

Statement refuted

Witness

Facts & Assumptions

Verification

Harmonic transmissibility across a one-dimensional material interface

Example

Facts & Assumptions

Verification

Affine anisotropic flux on a skew parallelogram

Example

Facts & Assumptions

Verification

Sources