Finite-volume diffusion on admissible meshes

Let $d\ge 1$ and let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected open polyhedral set, meaning the interior of a finite union of bounded closed convex polytopes. A closed convex polytope is a bounded intersection of finitely many closed half-spaces with nonempty interior. A cell-face mesh $\mathcal{T}=(\mathcal{K},\mathcal{F})$ consists of finitely many nonempty cells $K=\mathrm{int}{P}_K\subset \mathrm{\Omega }$, where each $P_K$ is a closed convex polytope, with pairwise disjoint interiors and $\bar{\mathrm{\Omega }}={\bigcup }_{K\in \mathcal{K}}\bar{K}$, together with the relative interiors of the $(d-1)$-dimensional faces of their closures. Each interior face $F\in {\mathcal{F}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ is incident to exactly two cells $K,L$ and is denoted $F=K\mathrm{\mid }L$; each boundary face $F\in {\mathcal{F}}_{\mathrm{b}\mathrm{d}\mathrm{y}}$ is incident to one cell and is contained in $\mathrm{\partial }\mathrm{\Omega }$. In dimension one a face is an endpoint and has measure $\mathrm{\mid }F\mathrm{\mid }=1$.

Write $\mathrm{\mid }K\mathrm{\mid }$ for the $d$-dimensional volume of $K$, $\mathrm{\mid }F\mathrm{\mid }$ for the $(d-1)$-dimensional measure of $F$, $n_{K,F}$ for the unit normal outward from $K$, $x_K$ for a chosen point in $K$, and $x_F$ for the centroid of $F$. For every incident pair set $d_{K,F}=\mathrm{dist}(x_K,F)>0$. The cell adjacency graph has one vertex for each cell and an edge for each interior face. A mesh is boundary anchored if every cell has a finite path in this graph to a cell incident to a boundary face; such a path can be chosen simple, with no repeated cells.

For a family $({\mathcal{T}}_h)$, put $h={\max }_{K\in \mathcal{K}}\mathrm{diam}K$. The family is uniformly regular if there are fixed constants $c,C>0$ and $N_0\in \mathbb{N}$ such that, for every mesh and each of its cells and faces, $c{h}^d\le \mathrm{\mid }K\mathrm{\mid }\le C{h}^d,c{h}^{d-1}\le \mathrm{\mid }F\mathrm{\mid }\le C{h}^{d-1},ch\le d_{K,F}\le Ch,$ and each cell has at most $N_0$ faces. A family is refining if $h\to 0$. These bounds are invoked only where explicitly assumed; a cell-face mesh alone is not asserted to be uniformly regular or orthogonal.

The finite-volume convention is that a face flux $Q_{K,F}$ is signed outward from $K$. If $F=K\mathrm{\mid }L$ is an interior face, local conservation requires the single shared-flux identity $Q_{K,F}+Q_{L,F}=0$. The mesh drawing below labels the center-to-center vector and the outward normal; no orthogonality is part of this general mesh definition.

The cell-face mesh definition and the diagram fix only the objects, incidences, normals, and orientation. A two-point flux approximation adds a separate orthogonality condition; the hybrid mimetic method is formulated on the general cell-face geometry specified here.

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.

Let $\mathcal{T}$ be a cell-face mesh as in Finite-volume cell-face meshes and geometric regularity, with $x_K$ the volume centroid of each cell and $x_F$ the face centroid. For the mimetic branch require the following explicit face-moment closure on each cell: ${\sum }_{F\subset \mathrm{\partial }K}\mathrm{\mid }F\mathrm{\mid }{n}_{K,F}=0,{\sum }_{F\subset \mathrm{\partial }K}\mathrm{\mid }F\mathrm{\mid }{x}_F\otimes n_{K,F}=\mathrm{\mid }K\mathrm{\mid }{I}_d\mathrm{.}$ Here $I_d$ is the identity matrix on ${\mathbb{R}}^d$ and $(x\otimes n{)}_{ij}=x_i{n}_j$. These are finite geometric checks on the face data; they impose no center-to-face orthogonality. For every cell use one real cell-mean unknown $u_K$; for every face use one shared real face-mean unknown $u_F$. On a boundary face $F\subset \mathrm{\partial }\mathrm{\Omega }$, the value $u_F=g_F$ is prescribed. Write $n_{K,F}$ for the outward unit normal. Define the reconstructed cell gradient and its face residual by $G_{K}u=\frac{1}{\mathrm{\mid }K\mathrm{\mid }}{\sum }_{F\subset \mathrm{\partial }K}\mathrm{\mid }F\mathrm{\mid }{u}_F{n}_{K,F},{\rho }_{K,F}(u)=u_F-u_K-G_{K}u\cdot (x_F-x_K)\mathrm{.}$ Interpret $u_K$ and $u_F$ as cell and face averages. For a symmetric positive-definite tensor $A_K$ constant on $K$ (so $z\cdot A_{K}z>0$ for every nonzero $z$), define ${\mathcal{A}}_K(u,v)=\mathrm{\mid }K\mathrm{\mid }(A_K{G}_{K}u)\cdot G_{K}v+{\lambda }_{\min }(A_K){\sum }_{F\subset \mathrm{\partial }K}\frac{\mathrm{\mid }F\mathrm{\mid }}{d_{K,F}}{\rho }_{K,F}(u){\rho }_{K,F}(v),$ where ${\lambda }_{\min }(A_K)>0$ is the smallest eigenvalue of $A_K$. For $f\in L^1(\mathrm{\Omega })$, the linear hybrid mimetic scheme with homogeneous Dirichlet data is to find all cell and interior-face values $u$ such that ${\sum }_{K\in \mathcal{K}}{\mathcal{A}}_K(u,v)={\sum }_{K\in \mathcal{K}}{v}_K{\int }_{K}f(x)dx$ for every test vector $v$ whose boundary face values are zero. For each local face $F\subset \mathrm{\partial }K$, let ${\varphi }_{K,F}$ have face value $1$ on $F$, zero on the other faces of $K$, and cell value zero. Define the outward numerical flux $Q_{K,F}(u)=-{\mathcal{A}}_K(u,{\varphi }_{K,F})$.

There is also a monotone $p$-growth backward-Euler extension, specified as an algebraic scheme but with no existence or convergence claim in this definition. Let $1<p<\mathrm{\infty }$, set $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$, and let $a(x,\xi )$ be a Carathéodory flux, meaning that $x\mapsto a(x,\xi )$ is measurable for each fixed $\xi$ and $\xi \mapsto a(x,\xi )$ is continuous for almost every $x$. Assume there are constants $C,\alpha >0$, a nonnegative $g\in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })$, and a nonnegative $h\in L^1(\mathrm{\Omega })$ such that, outside one null set, $\mathrm{\mid }a(x,\xi )\mathrm{\mid }\le C(g(x)+\mathrm{\mid }\xi {\mathrm{\mid }}^{p-1}),a(x,\xi )\cdot \xi \ge \alpha \mathrm{\mid }\xi {\mathrm{\mid }}^p-h(x),(a(x,\xi )-a(x,\eta ))\cdot (\xi -\eta )\ge 0.$ For each cell set $b_K(\xi )=\mathrm{\mid }K{\mathrm{\mid }}^{-1}{\int }_{K}a(x,\xi )dx$, and fix ${\alpha }_{0,K}>0$. Put ${\psi }_p(s)=\mathrm{\mid }s{\mathrm{\mid }}^{p-2}s$ for $s\mathrm{\ne }0$ and ${\psi }_p(0)=0$. Its spatial operator, linear in the test vector $v$, is ${\mathcal{A}}_{h,K}(u;v)=\mathrm{\mid }K\mathrm{\mid }{b}_K(G_{K}u)\cdot G_{K}v+{\alpha }_{0,K}{\sum }_{F\subset \mathrm{\partial }K}\frac{\mathrm{\mid }F\mathrm{\mid }}{d_{K,F}^{p-1}}{\psi }_p({\rho }_{K,F}(u)){\rho }_{K,F}(v)\mathrm{.}$ For this extension define $Q_{K,F}^{\mathrm{n}\mathrm{l}}(u)=-{\mathcal{A}}_{h,K}(u;{\varphi }_{K,F})$. On a time partition $0=t_0<\cdots <t_N=T$, given $u^{n-1}$ and $f\in L^{p^{\mathrm{\prime }}}((0,T)\times \mathrm{\Omega })$, put $f_K^n=((t_n-t_{n-1})\mathrm{\mid }K\mathrm{\mid }{)}^{-1}{\int }_{t_{n-1}}^{t_n}{\int }_{K}f$. The backward-Euler step seeks $u^n$ with zero boundary face values satisfying ${\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }\frac{u_K^n-u_K^{n-1}}{t_n-t_{n-1}}{v}_K+{\sum }_{K\in \mathcal{K}}{\mathcal{A}}_{h,K}(u^n;v)={\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }{f}_K^n{v}_K$ for every test vector with zero boundary face values. The formula specifies a candidate discrete method; solvability and convergence require separate results.

For an affine function $\mathrm{\ell }(x)=b\cdot x+c$, its cell and face averages satisfy $G_K\mathrm{\ell }=b$ and ${\rho }_{K,F}(\mathrm{\ell })=0$ by the displayed face-moment closure. The drawing illustrates the cell and face degrees of freedom on a skew cell; no center-to-face orthogonality is assumed.

When $p=2$, $b_K(\xi )=A_K\xi$, and ${\alpha }_{0,K}={\lambda }_{\min }(A_K)$, the nonlinear formula agrees with the linear form above. Neither specification by itself proves solvability, a conforming $W_0^{1,p}$ reconstruction, or a mesh-uniform discrete Aubin–Lions estimate; those require separate results or hypotheses.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz polyhedral domain, $d\ge 1$. A regular simplicial companion family consists of refining uniformly regular boundary-anchored cell-face meshes in Finite-volume cell-face meshes and geometric regularity, and, for each mesh, a matching simplicial subdivision ${\mathcal{S}}_h$ of its cell closures with the following finite geometric data and uniform constants $c,C,J>0$.

Each cell closure $\bar{K}$ is the union of at most $J$ closed nondegenerate $d$-simplices with pairwise disjoint interiors. Each simplex has diameter at most $Ch$ and inradius at least $ch$, and the chosen cell point $x_K$ is a subdivision vertex. The subdivision of each cell closure is a fan from $x_K$ over a triangulation of its boundary, and subdivisions agree on shared faces. Each original face closure $\bar{F}$ is a union of at most $J$ closed subdivision facets. In dimension one these mean the two subintervals from $x_K$ to the endpoints.

For each subdivision vertex $z$, let $\mathcal{K}(z)$ consist of the original cells whose closures contain $z$. This set has at most $J$ elements. For any $K,L\in \mathcal{K}(z)$ there is a path of at most $J$ cells in $\mathcal{K}(z)$, from $K$ to $L$, consecutive cells sharing an original face whose closure contains $z$. If $z\in \mathrm{\partial }\mathrm{\Omega }$, every $K\in \mathcal{K}(z)$ has such a path to a cell with a boundary face whose closure contains $z$. These are connectivity checks on finite vertex stars, not an assumption of an analytic inequality.

For shared face values $v_F$, cell values $v_K$, and homogeneous boundary values $v_F=0$, define the piecewise constant state ${\mathrm{\Pi }}_{h}v{\mathrm{\mid }}_K=v_K$ and $D_{h,p}(v{)}^p={\sum }_K{\sum }_{F\subset \mathrm{\partial }K}\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}^{1-p}\mathrm{\mid }{v}_F-v_K{\mathrm{\mid }}^p(1<p<\mathrm{\infty })\mathrm{.}$ The mean interpolator $I_{h}v$ has cell and face values equal to the corresponding means of a Sobolev function, with face means interpreted by the simplex trace; for $v\in W_0^{1,p}(\mathrm{\Omega })$ its boundary face means are zero. Define $E_{h}v$ to be continuous and affine on each subdivision simplex, with nodal value $\mathrm{\mid }\mathcal{K}(z){\mathrm{\mid }}^{-1}{\sum }_{K\in \mathcal{K}(z)}{v}_K$ at an interior vertex $z$ and zero at a boundary vertex. Thus $E_h$ is an explicit linear function of the cell values; its definition does not assume its stability.

These conditions allow nonorthogonal cells. A nonempty example in every dimension is the uniform cubical partition of $(0,1{)}^d$, with cell centers, a fixed matching barycentric subdivision of every face, and the cones from cell centers to those face simplices. Scaling leaves finitely many simplex shapes and bounded vertex stars; coordinate adjacency connects each interior star and connects each boundary star to a boundary face. In dimension one use equal subintervals and their midpoints. Hence all the displayed constants can be chosen independently of refinement.

Let $\mathcal{T}$ be a finite TPFA-admissible cell-face mesh, let every cell be connected through interior faces to a cell with a boundary face, and let $T_F>0$ be the transmissibilities of Orthogonal TPFA diffusion with harmonic face transmissibility. For cell vectors $u,v$, define $(A_{h}u{)}_K=\frac{1}{\mathrm{\mid }K\mathrm{\mid }}({\sum }_{F=K\mathrm{\mid }L}{T}_F(u_K-u_L)+{\sum }_{F\subset \mathrm{\partial }K\cap \mathrm{\partial }\mathrm{\Omega }}{T}_{K,F}{u}_K)\mathrm{.}$ Then interior fluxes are locally conservative, and for all cell vectors $u,v$, ${\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }{v}_K(A_{h}u{)}_K={\sum }_{F=K\mathrm{\mid }L\in {\mathcal{F}}_{\mathrm{i}\mathrm{n}\mathrm{t}}}{T}_F(v_K-v_L)(u_K-u_L)+{\sum }_{F\in {\mathcal{F}}_{\mathrm{b}\mathrm{d}\mathrm{y}}}{T}_{K,F}{v}_K{u}_K\mathrm{.}$ The stiffness matrix associated with the displayed bilinear identity is symmetric positive definite. Equivalently, $A_h$ is self-adjoint and positive definite in the volume-weighted inner product $⟨u,v{⟩}_h={\sum }_K\mathrm{\mid }K\mathrm{\mid }{u}_K{v}_K$; its ordinary coordinate matrix need not be symmetric when cell volumes differ. Hence every finite source and boundary-data vector gives a unique solution. If all cell sources and all prescribed boundary face values are nonnegative, then every solution cell value is nonnegative. The maximum-principle claim uses the boundary anchor; no diagonal sign statement alone is used as a substitute for that condition.

Let $\mathcal{T}$ be a finite boundary-anchored TPFA mesh and let $T_e>0$ be its interior and boundary transmissibilities. A boundary path family assigns to each cell $K$ a finite simple path ${\gamma }_K$ of adjacent interior faces, followed by one boundary face of its terminal cell; simple means it visits no cell more than once. Set ${\mathrm{\Delta }}_{e}u=u_K-u_L$ on an interior face $e=K\mathrm{\mid }L$ with a fixed orientation, and set ${\mathrm{\Delta }}_{e}u=u_K$ on a boundary face of $K$. The sign of an interior increment is reversed when the path traverses the fixed orientation backward. Put $S_K={\sum }_{e\in {\gamma }_K}{T}_e^{-1},C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}={\max }_e{\sum }_{K:e\in {\gamma }_K}\mathrm{\mid }K\mathrm{\mid }{S}_K\mathrm{.}$ Then every cell vector $u$ satisfies the discrete Poincaré inequality ${\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }{u}_K^2\le C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}{\sum }_e{T}_e({\mathrm{\Delta }}_{e}u{)}^2=C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}\mathrm{\parallel }u{\mathrm{\parallel }}_{E_h}^2\mathrm{.}$ Consequently, any uniformly regular mesh family supplied with boundary path families whose $C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ are uniformly bounded has a mesh-uniform Poincaré constant. This theorem does not assert that every orthogonal mesh family has such a uniform path bound.

Let a finite boundary-anchored cell-face mesh satisfy the face-moment closure in Hybrid mimetic finite-volume diffusion on general meshes, and let every $A_K$ be symmetric positive definite. Define the local seminorm $\mathrm{\parallel }v{\mathrm{\parallel }}_{K,\mathrm{m}\mathrm{i}\mathrm{m}}^2=\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{G}_{K}v{\mathrm{\mid }}^2+{\sum }_{F\subset \mathrm{\partial }K}\frac{\mathrm{\mid }F\mathrm{\mid }}{d_{K,F}}{\rho }_{K,F}(v{)}^2\mathrm{.}$ Its kernel consists exactly of the constant local vectors $v_K=v_F=c$, so it induces a norm modulo constants. Then ${\mathcal{A}}_K(v,v)\ge {\lambda }_{\min }(A_K)\mathrm{\parallel }v{\mathrm{\parallel }}_{K,\mathrm{m}\mathrm{i}\mathrm{m}}^2$. The global Dirichlet matrix of the linear hybrid mimetic scheme is symmetric positive definite and has a unique solution for every finite cell-source vector. Its induced numerical fluxes satisfy the cellwise balance and opposite-flux identity on each interior face. For any affine $\mathrm{\ell }(x)=b\cdot x+c$, the local linear flux is exactly $Q_{K,F}(\mathrm{\ell })=-\mathrm{\mid }F\mathrm{\mid }(A_{K}b)\cdot n_{K,F}$. For the nonlinear $p$-growth extension in Hybrid mimetic finite-volume diffusion on general meshes, the spatial operator is monotone and coercive in the reconstructed gradient and residual variables, and its shared interior face fluxes are conservative at each implicit time step.

The proposition proves local coercivity in the explicitly defined gradient-residual seminorm (a norm modulo local constants) and affine flux exactness. It does not assert a discrete maximum principle for the mimetic scheme or a mesh-uniform condition number without additional mesh-family estimates.

Assume the Axiom of Choice and a family in Finite-volume families with a regular simplicial companion. For $1<p<\mathrm{\infty }$ and zero boundary face values, put $B_{h,p}(v{)}^p={\sum }_K\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }{G}_{K}v{\mathrm{\mid }}^p+S_{h,p}(v{)}^p,S_{h,p}(v{)}^p={\sum }_{K,F}\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}^{1-p}\mathrm{\mid }{\rho }_{K,F}(v){\mathrm{\mid }}^p,$ where $G_K,{\rho }_{K,F}$ have the formulas in Hybrid mimetic finite-volume diffusion on general meshes. Then, with constants independent of $h,v$, $C^{-1}{D}_{h,p}(v)\le B_{h,p}(v)\le C{D}_{h,p}(v),\mathrm{\parallel }{E}_{h}v{\mathrm{\parallel }}_{W^{1,p}}\le C{D}_{h,p}(v),\mathrm{\parallel }{\mathrm{\Pi }}_{h}v-E_{h}v{\mathrm{\parallel }}_{L^p}\le Ch{D}_{h,p}(v)\mathrm{.}$ Here $E_{h}v\in W_0^{1,p}(\mathrm{\Omega })$. If $p>\max (1,2d\mathrm{/}(d+2))$ and $\vartheta =1-d\max (1\mathrm{/}p-1\mathrm{/}2,0)>0$, also $\mathrm{\parallel }{\mathrm{\Pi }}_{h}v-E_{h}v{\mathrm{\parallel }}_2\le C{h}^{\vartheta }{D}_{h,p}(v),\mathrm{\parallel }{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_2+\mathrm{\parallel }{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_p\le C{D}_{h,p}(v)\mathrm{.}$ For $z\in W_0^{1,p}(\mathrm{\Omega })$, $D_{h,p}(I_{h}z)\le C\mathrm{\parallel }\mathrm{\nabla }z{\mathrm{\parallel }}_p$. If the cell points are volume centroids, then for smooth $z$ with compact support in $\mathrm{\Omega }$, $G_h{I}_{h}z\to \mathrm{\nabla }z$ uniformly and $S_{h,p}(I_{h}z)\le Ch\mathrm{\parallel }{D}^{2}z{\mathrm{\parallel }}_{\mathrm{\infty }}$, where $G_{h}v{\mathrm{\mid }}_K=G_{K}v$. Finally, for every smooth vector field $\varphi$ on a neighborhood of $\bar{\mathrm{\Omega }}$, $\mid {\int }_{\mathrm{\Omega }}{G}_{h}v\cdot \varphi +{\mathrm{\Pi }}_{h}v\mathrm{div}\varphi \mid \le Ch{D}_{h,p}(v)\mathrm{\parallel }\mathrm{\nabla }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ Consequently bounded $D_{h,p}(v_h)$, strong $L^p$ convergence ${\mathrm{\Pi }}_h{v}_h\to v$, and weak $L^p$ convergence $G_h{v}_h\rightharpoonup g$ imply $g=\mathrm{\nabla }v$ distributionally. The same statements hold after integration in time with the corresponding $L^p$ bounds.