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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 $K$ .

Depends on

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Sources

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, Vol. VII, 2000; complete updated author text (standard reference, not scraped)