No-flux TPFA diffusion generator

Let $\mathcal{T}=(\mathcal{K},\mathcal{F})$ be a finite TPFA-admissible cell-face mesh from Finite-volume cell-face meshes and geometric regularity and Orthogonal TPFA diffusion with harmonic face transmissibility. Impose homogeneous no-flux boundary conditions: boundary faces contribute zero flux and therefore no conductance. For adjacent cells $K,L$, set

$$c_{KL}=\sum _{F=K\mathrm{\mid }L}{T}_F,$$

where the sum is over their common interior faces and is zero when they are not adjacent. Set $c_{KL}=c_{LK}$ and $m_K=\mathrm{\mid }K\mathrm{\mid }$. If more than one face separates a cell pair, summing their conductances gives the same cell-balance flux as retaining the faces separately. Define the symmetric matrix $L_h$ by $(L_h{)}_{KK}={\sum }_{L\mathrm{\ne }K}{c}_{KL}$ and $(L_h{)}_{KL}=-c_{KL}$ for $K\mathrm{\ne }L$.

For species diffusivities $d_r\ge 0$, the no-flux finite-volume diffusion equation is $M{U}_r^{\mathrm{\prime }}+d_r{L}_h{U}_r=0$ for each species $r$, where $M=\mathrm{diag}(\mathrm{\mid }K\mathrm{\mid })$. Its component equation is

$$\mathrm{\mid }K\mathrm{\mid }(U_{K,r}{)}^{\mathrm{\prime }}=d_r\sum _{L:c_{KL}>0}{c}_{KL}(U_{L,r}-U_{K,r})\mathrm{.}$$

The equality is the usual shared-face TPFA balance with outward flux $T_F(U_{K,r}-U_{L,r})$ and no boundary flux. The matrix $L_h$ is symmetric, has zero row sums and nonpositive off-diagonal entries, and obeys $v^{\mathsf{T}}{L}_{h}v={\sum }_{\{K,L\}}{c}_{KL}(v_K-v_L{)}^2\ge 0$, where the sum is over unordered adjacent cell pairs. With $M=\mathrm{diag}(\mathrm{\mid }K\mathrm{\mid })$, the generator $-M^{-1}{L}_h$ is self-adjoint in the weighted pairing $⟨v,w{⟩}_M=v^{\mathsf{T}}Mw$. This definition makes no accuracy claim for arbitrary admissible meshes.