Hybrid mimetic finite-volume diffusion on general meshes

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.