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TPFA discrete Green identity, coercivity, and maximum principle

Statement

Let $T$ be a finite TPFA-admissible cell-face mesh, let every cell be connected through interior faces to a cell with a boundary face, and let $T_{F} > 0$ be the transmissibilities of Orthogonal TPFA diffusion with harmonic face transmissibility. For cell vectors $u, v$ , define $(A_{h} u)_{K} = \frac{1}{∣ K ∣} (\sum_{F = K ∣ L} T_{F} (u_{K} - u_{L}) + \sum_{F \subset \partial K \cap \partial Ω} T_{K, F} u_{K}) .$ Then interior fluxes are locally conservative, and for all cell vectors $u, v$ , $\sum_{K \in K} ∣ K ∣ v_{K} (A_{h} u)_{K} = \sum_{F = K ∣ L \in F_{i n t}} T_{F} (v_{K} - v_{L}) (u_{K} - u_{L}) + \sum_{F \in F_{b d y}} T_{K, F} v_{K} u_{K} .$ The stiffness matrix associated with the displayed bilinear identity is symmetric positive definite. Equivalently, $A_{h}$ is self-adjoint and positive definite in the volume-weighted inner product $⟨ u, v ⟩_{h} = \sum_{K} ∣ K ∣ u_{K} v_{K}$ ; its ordinary coordinate matrix need not be symmetric when cell volumes differ. Hence every finite source and boundary-data vector gives a unique solution. If all cell sources and all prescribed boundary face values are nonnegative, then every solution cell value is nonnegative. The maximum-principle claim uses the boundary anchor; no diagonal sign statement alone is used as a substitute for that condition.

Facts & Assumptions

Given: A finite connected cell-face mesh, the positive TPFA transmissibilities, a path from every cell to a boundary face, finite real cell and face vectors, and the cell-balance convention of Orthogonal TPFA diffusion with harmonic face transmissibility.

[F1]

Finite-volume cell-face meshes and geometric regularity: the cell adjacency graph is finite; an interior face belongs to exactly two cells, and a boundary face belongs to one cell.

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: $T_{F} > 0$ ; the two outward fluxes on an interior face are $T_{F} (u_{K} - u_{L})$ and $T_{F} (u_{L} - u_{K})$ ; a homogeneous boundary face contributes $T_{K, F} u_{K}$ to the cell operator.

[A1]

The mesh is boundary anchored: every cell has a finite adjacency path to a cell incident to a Dirichlet boundary face. All cell volumes and transmissibilities are strictly positive.

Proof

1.1F1F2algebra

For an interior face $F = K ∣ L$ , the shared-flux definitions give $Q_{K, F} (u) + Q_{L, F} (u) = T_{F} (u_{K} - u_{L}) + T_{F} (u_{L} - u_{K}) = 0$ . Thus summing cell balances cancels every interior flux exactly.

1.2F1F2givenalgebra

In $\sum_{K} ∣ K ∣ v_{K} (A_{h} u)_{K}$ , the two terms contributed by an interior face are $T_{F} [v_{K} (u_{K} - u_{L}) + v_{L} (u_{L} - u_{K})] = T_{F} (v_{K} - v_{L}) (u_{K} - u_{L})$ . A boundary face contributes $T_{K, F} v_{K} u_{K}$ . Summing all faces proves the displayed identity.

2.1A1step 1.2algebra

Set $v = u$ in step 1.2. The resulting stiffness quadratic form is a sum of nonnegative terms. If it is zero, every interior term forces $u_{K} = u_{L}$ , and every boundary term forces $u_{K} = 0$ on its incident cell. The path condition [A1] propagates these zeros through the connected cell graph, so $u = 0$ . The finite-dimensional symmetric stiffness matrix therefore has trivial kernel and is positive definite and invertible. Since the volume matrix $M_{h} = diag (∣ K ∣)$ is invertible, $A_{h} = M_{h}^{- 1} (M_{h} A_{h})$ is also invertible; inhomogeneous boundary values change only the right-hand side.

3.1F1F2A1choosealgebra∎

Suppose $u$ solves the cell balances with $f_{K} \geq 0$ and $g_{F} \geq 0$ . If a cell value has a negative global minimum, choose a minimizing cell $K$ . Every interior summand $T_{F} (u_{K} - u_{L})$ in its balance is nonpositive, and every boundary summand $T_{K, F} (u_{K} - g_{F})$ is strictly negative. Since the balance equals $∣ K ∣ f_{K} \geq 0$ , all summands must be zero. Thus the minimum propagates unchanged to every adjacent cell along a path from [A1], but at a boundary face the summand $T_{K, F} (u_{K} - g_{F})$ is strictly negative. This contradiction rules out a negative minimum, so all cell values are nonnegative.

Depends on

Used by

TPFA flux-energy convergence for smooth face-aligned interfaces Theorem

Dependency tree · one level

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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)
Jérôme Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods, 2014 (standard reference, not scraped)