No-flux finite-volume and conforming realizations for phase fields

The no-flux finite-volume realization starts from a finite connected cell-face mesh as in Finite-volume cell-face meshes and geometric regularity and uses the interior-face orthogonality and positive transmissibilities of Orthogonal TPFA diffusion with harmonic face transmissibility. For unit diffusivity, let $X_h={\mathbb{R}}^{\mathcal{K}}$, $m_h(u,v)={\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }{u}_K{v}_K$, and $a_h(u,v)={\sum }_{F=K\mathrm{\mid }L\in {\mathcal{F}}_{\mathrm{i}\mathrm{n}\mathrm{t}}}{T}_F(u_K-u_L)(v_K-v_L)\mathrm{.}$ There is no boundary-face term: the flux through each physical boundary face is prescribed to be zero. Define $A_h:X_h\to X_h$ by $m_h(A_{h}u,v)=a_h(u,v)$ for all $v\in X_h$. Split the double-well density as $F_{\epsilon }=F_c-F_e,F_c(s)=\frac{s^4}{4{\epsilon }^2}+\frac{s^2}{2{\epsilon }^2},F_e(s)=\frac{s^2}{{\epsilon }^2}-\frac{1}{4{\epsilon }^2}\mathrm{.}$ The associated finite-volume energy components are $E_{c,h}(u)={\sum }_K\mathrm{\mid }K\mathrm{\mid }(\frac{u_K^4}{4{\epsilon }^2}+\frac{u_K^2}{2{\epsilon }^2}),E_{e,h}(u)={\sum }_K\mathrm{\mid }K\mathrm{\mid }(\frac{u_K^2}{{\epsilon }^2}-\frac{1}{4{\epsilon }^2})\mathrm{.}$

For convergence on $\mathrm{\Omega }=(0,L)$, a second realization uses the same orthogonal dual mesh with centers $x_i=(i-\frac{1}{2})h$, $h=L\mathrm{/}N$. Its conforming space $V_h$ consists of functions affine between consecutive centers and constant on the two half-cells between a boundary and the nearest center. Its inner product is the exact $L^2(0,L)$ inner product, its stiffness is $a_h(v,w)={\int }_0^L{v}^{\mathrm{\prime }}{w}^{\mathrm{\prime }}dx$, and its energy components are $E_{c,h}(v)={\int }_0^L{F}_c(v)dx$ and $E_{e,h}(v)={\int }_0^L{F}_e(v)dx$. Define $A_h:V_h\to V_h$ by $(A_{h}v,w{)}_{L^2}=a_h(v,w)$. For degrees of freedom $v_i=v(x_i)$, this stiffness is exactly the interior two-point flux form ${\sum }_{i=1}^{N-1}(v_{i+1}-v_i)(w_{i+1}-w_i)\mathrm{/}h$; the exact mass and potential integrals are the conforming reconstruction used in the convergence theorem.

Both realizations have a constant function $\mathbf{1}$, a real finite-dimensional inner-product space, and a self-adjoint nonnegative operator $A_h$. They share the physical energy $E_h(u)=\frac{\kappa }{2}{m}_h(A_{h}u,u)+E_{c,h}(u)-E_{e,h}(u)$, with the appropriate inner product and energy components for the realization. The sign convention is $A_h\approx -\mathrm{\Delta }$.

The diagram labels four adjacent cell values, three shared interior fluxes, and the prescribed zero boundary fluxes. Each interior flux appears with opposite signs in its two cell balances. The diagram records the convention only; it is not used as a proof.

The published TPFA theorem TPFA discrete Green identity, coercivity, and maximum principle treats homogeneous Dirichlet data and its energy includes boundary penalties. The no-flux realization above deliberately omits those terms and uses only its interior transmissibilities; its kernel and gauge are proved separately in Kernel, range, and mean-zero inverse for the no-flux Laplacian. The conditional parabolic limit item Conditional monotone finite-volume parabolic limit identification assumes zero-boundary $W_0^{1,p}$ reconstructions and is not used for this Neumann branch. The finite-dimensional gradient-flow algorithm also applies to the conforming realization stated above.