Cell-centered rectangular TPFA grid with homogeneous no-flux boundaries

Let $\mathrm{\Omega }={\prod }_{j=1}^d(0,L_j)$, where $L_j>0$, and let integers $N_j\ge 2$ define $h_j=L_j\mathrm{/}{N}_j$. The finite cell set is $I_h={\prod }_{j=1}^d\{0,\dots ,N_j-1\}$, cell centers are $x_i=((i_j+\frac{1}{2})h_j{)}_{j=1}^d$, and every cell has volume $m_h={\prod }_j{h}_j$. This is the uniform orthogonal Cartesian mesh; adjacent center segments are perpendicular to their shared faces. Take the scalar cellwise coefficient to be ${\lambda }_K=1$ and give boundary faces homogeneous no-flux data, so they contribute no flux.

For a grid vector $v=(v_i{)}_{i\in I_h}$, define the positive cell-centered stencil

$$(A_h^{N}v{)}_i=\sum _{j=1}^d\frac{2v_i-v_{i+e_j}-v_{i-e_j}}{h_j^2},$$

where an unavailable neighbor at a boundary is assigned the value of the boundary cell itself. Indeed a face normal to direction $j$ has area $\mathrm{\mid }F\mathrm{\mid }=m_h\mathrm{/}{h}_j$ and adjacent centers are distance $h_j$ apart, so its TPFA transmissibility is $\mathrm{\mid }F\mathrm{\mid }\mathrm{/}{h}_j=m_h\mathrm{/}{h}_j^2$; dividing the cell balance by $m_h$ gives the displayed coefficient. Boundary faces contribute zero flux. Thus $M_h=m_{h}I$ and $L_h^N=m_h{A}_h^N$ are the mass and stiffness matrices of the no-flux specialization in No-flux TPFA diffusion generator. For species diffusivities $d_r\ge 0$, the finite-volume diffusion equation is $U_r^{\mathrm{\prime }}=-d_r{A}_h^N{U}_r$.

Define the doubled periodic grid with $2N_j$ cells per direction on side lengths $2L_j$, using the same widths $h_j$, as in Periodic rectangular reaction-diffusion grid. The reflection map $E_h$ extends $U$ evenly across every boundary face: in one coordinate, its extended index is $i$ for $0\le i<N_j$ and $2N_j-1-i$ for $N_j\le i<2N_j$; apply this coordinatewise. This finite reflection convention is used in the error theorem below.