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TPFA flux-energy convergence for smooth face-aligned interfaces

Statement

Let $Ω \subset R^{d}$ be a bounded connected polyhedral set and let $(T_{h})$ be a refining, uniformly regular, boundary-anchored family of TPFA-admissible meshes. Assume $λ_{K} \in [λ_{*}, λ^{*}]$ with $0 < λ_{*} \leq λ^{*} < \infty$ , constant on each cell, and that coefficient jumps occur only across mesh faces. Suppose $u$ is continuous, vanishes on $\partial Ω$ , and is $C^{2}$ on each cell closure with the uniform family bound $M = \sup_{h} \max_{K \in K_{h}} ∥ D^{2} u ∥_{L^{\infty} (K)} < \infty .$ For every interior face $F = K ∣ L$ fix an orientation and define its exact outward flux from $K$ by $J_{K, F} = - \int_{F} λ_{K} \nabla u ∣_{K} \cdot n_{K, F}$ . On a boundary face define $J_{K, F}$ by the same formula. Assume the exact fluxes satisfy the cell balances $\sum_{F \subset \partial K} J_{K, F} = \int_{K} f$ and $J_{K, F} = - J_{L, F}$ on every interior face. Let $U_{h}$ solve the TPFA cell balances with source $f$ and zero boundary values, and let $I_{h} u$ be the cell-center sample $u (x_{K})$ with zero boundary face values. Then, for a constant $C$ depending only on $Ω$ , $λ_{*}, λ^{*}$ , and the uniform mesh regularity bounds, $∥ U_{h} - I_{h} u ∥_{E_{h}} \leq C h M, {(\sum_{F \in F_{i n t}} \frac{∣ Q_{K, F} (U_{h}) - J_{K, F} ∣^{2}}{T_{F}} + \sum_{F \in F_{b d y}} \frac{∣ Q_{K, F} (U_{h}) - J_{K, F} ∣^{2}}{T_{K, F}})}^{1 / 2} \leq 2 C h M .$ The second estimate is convergence of the reconstructed numerical face flux in the transmissibility-dual energy norm. No pointwise operator consistency on arbitrary nonorthogonal meshes or error rate for nonsmooth weak solutions is asserted.

Facts & Assumptions

Given: The exact cell and face geometry, a refining uniformly regular orthogonal mesh family, $λ_{*} \leq λ_{K} \leq λ^{*}$ with mesh-aligned jumps, the piecewise $C^{2}$ function $u$ , the exact face fluxes and their cell balances, and the TPFA scheme.

[F1]

Finite-volume cell-face meshes and geometric regularity: in a uniformly regular family, cell volumes and face measures scale as $h^{d}$ and $h^{d - 1}$ , center-to-face distances scale as $h$ , and the number of faces per cell is uniformly bounded.

[F2]

Orthogonal TPFA diffusion with harmonic face transmissibility: $T_{F} = ∣ F ∣ / (d_{K, F} / λ_{K} + d_{L, F} / λ_{L})$ on interior faces, $T_{K, F} = ∣ F ∣ λ_{K} / d_{K, F}$ on boundary faces, and $∥ v ∥_{E_{h}}^{2} = \sum_{F} T_{F} (Δ_{F} v)^{2}$ .

[F3]

TPFA discrete Green identity, coercivity, and maximum principle: the TPFA stiffness matrix is symmetric positive definite, and its bilinear form is the face energy in [F2].

[A1]

Each cell restriction of $u$ extends $C^{2}$ to its closure with Hessian norm at most the family-wide bound $M$ . The exact integrated fluxes are opposite on each interior face, and the exact cell flux balances hold as stated.

Proof

1.1F1F2A1givenalgebra

For each face fix an incident cell $K$ and set $δ_{F} = Q_{K, F} (I_{h} u) - J_{K, F}$ . On an interior face $F = K ∣ L$ , write $n = n_{K, F}$ , let $y_{F}$ be the center-normal crossing point, and put $s_{F} = - J_{K, F} / ∣ F ∣$ . Integrated conservation says that $s_{F}$ is the face mean of both $λ_{K} \partial_{n} u_{K}$ and $λ_{L} \partial_{n} u_{L}$ . On each cell closure the Hessian bound gives $∣ λ_{K} \partial_{n} u_{K} (y_{F}) - s_{F} ∣ \leq C λ^{*} M h$ , and the same bound holds for $L$ . Continuity of $u$ and one-sided normal Taylor expansion therefore give $u (x_{K}) = u (y_{F}) - d_{K, F} s_{F} / λ_{K} + R_{K}$ and $u (x_{L}) = u (y_{F}) + d_{L, F} s_{F} / λ_{L} + R_{L}$ , with $∣ R_{K} ∣ + ∣ R_{L} ∣ \leq C M h^{2}$ . Substitution into the harmonic formula yields $Q_{K, F} (I_{h} u) = - ∣ F ∣ s_{F} + T_{F} (R_{K} - R_{L})$ , while $J_{K, F} = - ∣ F ∣ s_{F}$ . Since $T_{F} \leq C ∣ F ∣ / h$ , $∣ δ_{F} ∣ \leq C M ∣ F ∣ h$ . On a boundary face take $s_{F} = - J_{K, F} / ∣ F ∣$ and $y_{F} = x_{F}$ . The same face-mean estimate and $u (y_{F}) = 0$ give $u (x_{K}) = - d_{K, F} s_{F} / λ_{K} + R_{K}$ , $∣ R_{K} ∣ \leq C M h^{2}$ , and the identical defect bound.

2.1F1F2step 1.1algebra

By uniform regularity, every transmissibility satisfies $T_{F} \geq c ∣ F ∣ / h$ . There are at most $N_{0}$ faces per cell, $∣ F ∣ \leq C h^{d - 1}$ , and $∣ K ∣ \geq c h^{d}$ , so $# K_{h} \leq ∣ Ω ∣ / (c h^{d})$ and $\sum_{F} ∣ F ∣ \leq C h^{- 1}$ . Step 1.1 therefore gives $\sum_{F} \frac{δ_{F}^{2}}{T_{F}} \leq C M^{2} h^{3} \sum_{F} ∣ F ∣ \leq C M^{2} h^{2} .$ Thus the vector of local flux defects has transmissibility-dual norm at most $C h M$ .

3.1A1F3step 2.1algebra

Let $e = U_{h} - I_{h} u$ . The exact cell balances in [A1] and the finite-volume balances for $U_{h}$ imply, for every cell vector $v$ , $a_{h} (e, v) = - \sum_{F \in F_{i n t}} δ_{F} (v_{K} - v_{L}) - \sum_{F \in F_{b d y}} δ_{F} v_{K},$ because the two exact interior fluxes are oppositely oriented. Take $v = e$ . Weighted Cauchy–Schwarz gives $∥ e ∥_{E_{h}}^{2} \leq (\sum_{F} δ_{F}^{2} / T_{F})^{1 / 2} ∥ e ∥_{E_{h}}$ . If $e = 0$ the first estimate is immediate; otherwise divide by $∥ e ∥_{E_{h}}$ and use step 2.1 to obtain $∥ e ∥_{E_{h}} \leq C h M$ . The bilinear identity used here is [F3].

4.1F2step 2.1step 3.1algebra∎

On every face, $Q_{K, F} (U_{h}) - Q_{K, F} (I_{h} u)$ equals $T_{F} (e_{K} - e_{L})$ for an interior face and $T_{K, F} e_{K}$ for a boundary face. Its transmissibility-dual norm is therefore exactly $∥ e ∥_{E_{h}}$ . By the triangle inequality, the numerical-flux error norm is at most $∥ e ∥_{E_{h}} + (\sum_{F} δ_{F}^{2} / T_{F})^{1 / 2} \leq 2 C h M$ , using steps 2.1 and 3.1. This proves both estimates.

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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)