Hybridized no-flow RT0/P0 Darcy system

Assume Countable Choice.

Assume the no-flow Darcy data of No-flow mixed Darcy problem and coefficient norms and a facet-connected conforming mesh from Affine simplicial meshes and RT/BDM spaces. Let ${\mathcal{F}}_h^i$ be its interior facets. On each cell $K$, let $W_K:=\{v\in {\mathrm{RT}}_0(K):v\cdot n_K=0\text{ on every boundary facet of }D\text{ contained in }\mathrm{\partial }K\}\mathrm{.}$ The broken flux space is $W_h:={\prod }_{K\in {\mathcal{T}}_h}{W}_K$, the cell pressure space is $M_h:={\prod }_{K\in {\mathcal{T}}_h}\mathbb{R}$, and the skeleton multiplier space is ${\mathrm{\Lambda }}_h:={\prod }_{F\in {\mathcal{F}}_h^i}\mathbb{R}$. The scalar ${\lambda }_F$ is constant on facet $F$ and uses the fixed orientation only to label that facet; local equations use the outward normal of each cell.

For the source-only problem $F=0$, a hybrid solution $(u_h,p_h,{\lambda }_h)\in W_h\times M_h\times {\mathrm{\Lambda }}_h$ for $f\in Q$ satisfies, for every cell $K$, every $v_K\in W_K$, and every interior facet $F=K^{+}\cap K^{-}$, ${\int }_K{K}^{-1}{u}_K\cdot v_K-{\int }_K{p}_K\mathrm{div}{v}_K+{\sum }_{F\subset \mathrm{\partial }K,F\in {\mathcal{F}}_h^i}{\lambda }_F{\int }_F{v}_K\cdot n_K=0,$ ${\int }_K\mathrm{div}{u}_K={\int }_{K}f,$ ${\int }_F(u_{K^{+}}\cdot n_{K^{+}}+u_{K^{-}}\cdot n_{K^{-}})=0.$ Because each RT0 normal component is constant on a facet, the last equation is pointwise normal-flux matching. With physical-boundary flux zero, the transformation $p_K\mapsto p_K+c$ for all cells and ${\lambda }_F\mapsto {\lambda }_F+c$ leaves the hybrid equations unchanged: in the local first equation the change is $-c{\int }_K\mathrm{div}{v}_K+c{\sum }_{F\text{ interior}}{\int }_F{v}_K\cdot n_K=0$ by Gauss–Green and the zero physical-boundary flux; the mass and jump equations do not use pressure or multipliers. This is the common pressure gauge.