Orthogonal TPFA diffusion with harmonic face transmissibility

Let $\mathcal{T}=(\mathcal{K},\mathcal{F})$ be a boundary-anchored mesh as in Finite-volume cell-face meshes and geometric regularity. It is TPFA-admissible if, for each interior face $F=K\mathrm{\mid }L$, the center segment $[x_K,x_L]$ is perpendicular to the supporting hyperplane of $F$ and intersects the relative interior of $F$ at a point $y_F$. Thus the center points lie on opposite sides of that face, and $d_{K,F}=\mathrm{\mid }{y}_F-x_K\mathrm{\mid }$, $d_{L,F}=\mathrm{\mid }{x}_L-y_F\mathrm{\mid }$, so define $d_F:=\mathrm{\mid }{x}_L-x_K\mathrm{\mid }=d_{K,F}+d_{L,F}$ for every interior face. Each boundary face $F\subset \mathrm{\partial }K\cap \mathrm{\partial }\mathrm{\Omega }$ must satisfy $x_F-x_K=d_{K,F}{n}_{K,F}$ with $d_{K,F}>0$; in particular, its face centroid is the normal projection of $x_K$ onto the face plane.

Let the scalar diffusion coefficient be constant on each cell, $\lambda {\mathrm{\mid }}_K={\lambda }_K$, with fixed bounds $0<{\lambda }_{\ast }\le {\lambda }_K\le {\lambda }^{\ast }<\mathrm{\infty }$. For each interior face and boundary face define the transmissibilities $T_F=\frac{\mathrm{\mid }F\mathrm{\mid }}{d_{K,F}\mathrm{/}{\lambda }_K+d_{L,F}\mathrm{/}{\lambda }_L}(F=K\mathrm{\mid }L),T_{K,F}=\frac{\mathrm{\mid }F\mathrm{\mid }{\lambda }_K}{d_{K,F}}(F\subset \mathrm{\partial }K\cap \mathrm{\partial }\mathrm{\Omega })\mathrm{.}$ The interior formula is the conductance of two half-cell resistances in series. Equivalently, with ${\lambda }_F^{\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}}=d_F\mathrm{/}(d_{K,F}\mathrm{/}{\lambda }_K+d_{L,F}\mathrm{/}{\lambda }_L)$, it is $T_F=\mathrm{\mid }F\mathrm{\mid }{\lambda }_F^{\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}}\mathrm{/}{d}_F$. This is a distance-weighted harmonic mean; it equals the ordinary harmonic mean only when $d_{K,F}=d_{L,F}$.

For cell values $u_K$ and prescribed boundary face values $g_F$, the outward signed flux is $Q_{K,F}(u)=T_F(u_K-u_L)(F=K\mathrm{\mid }L),Q_{K,F}(u)=T_{K,F}(u_K-g_F)(F\subset \mathrm{\partial }\mathrm{\Omega })\mathrm{.}$ This is a two-point flux approximation (TPFA): an interior face flux uses only the values in its two incident cells, and a boundary face flux uses only its incident cell value and the prescribed boundary value. The finite-volume equations for a cellwise integrable source $f$ are ${\sum }_{F\subset \mathrm{\partial }K}{Q}_{K,F}(u)={\int }_{K}f(x)dx(K\in \mathcal{K})\mathrm{.}$ Thus an interior face uses one flux with opposite signs in its two balances. For homogeneous Dirichlet data define the symmetric energy form and its norm by $a_h(u,v)={\sum }_{F=K\mathrm{\mid }L\in {\mathcal{F}}_{\mathrm{i}\mathrm{n}\mathrm{t}}}{T}_F(u_K-u_L)(v_K-v_L)+{\sum }_{F\in {\mathcal{F}}_{\mathrm{b}\mathrm{d}\mathrm{y}}}{T}_{K,F}{u}_K{v}_K,\mathrm{\parallel }u{\mathrm{\parallel }}_{E_h}^2=a_h(u,u)\mathrm{.}$ For each face take a separate copy $D_F$ of measure $\mathrm{\mid }F\mathrm{\mid }{d}_F$ on an interior face and $\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}$ on a boundary face. The reconstructed normal gradient is $(u_L-u_K)n_{K,F}\mathrm{/}{d}_F$ on an interior face (with either fixed orientation) and $-u_K{n}_{K,F}\mathrm{/}{d}_{K,F}$ on a homogeneous boundary face. Give the interior face copy coefficient ${\lambda }_F^{\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}}$ and the boundary face copy coefficient ${\lambda }_K$. Then $\mathrm{\parallel }u{\mathrm{\parallel }}_{E_h}^2$ is exactly the weighted $L^2$ norm of this face-copy gradient reconstruction. The copies are bookkeeping spaces; they are not claimed to partition $\mathrm{\Omega }$.

The drawing uses the outward-from-cell sign convention, so $Q_{K,F}=-Q_{L,F}$. Orthogonality is required for the TPFA diffusion flux to approximate the full normal flux with only the two cell values. The definition includes scalar isotropic diffusion only; anisotropic tensors and nonorthogonal meshes require a separately analysed cell-and-face method. Cellwise coefficient jumps are allowed when their interfaces are mesh faces. No universal consistency claim is made for jumps that cut through cells.