Hybridization, static condensation, and pressure gauge

Start with a hybrid solution. Its jump constraints and zero boundary flux, together with [F2], make $u_h$ a member of $V_h$. Summing the cell balances against any cellwise constant mean-zero $q_h$ gives the discrete second mixed equation.

Conversely let $(u_h,p_h)$ be the unique conforming solution. Its cell balances hold by [F4]. Let $W_h$ be the broken flux space and define the jump map $J:W_h\to {\mathbb{R}}^{{\mathcal{F}}_h^i}$ by $J_F(v)={\int }_F(v_{K^{+}}\cdot n_{K^{+}}+v_{K^{-}}\cdot n_{K^{-}})$. Its kernel consists of fields with matching interior fluxes, hence lies in $V_h$ by [F2]. Define $\mathrm{\ell }(v)={\sum }_K[a_K(u_h,v_K)-{\int }_K{p}_h\mathrm{div}{v}_K]$. The conforming first equation gives $\mathrm{\ell }=0$ on $\ker J$.

Suppose there are at least two cells. Every cell is incident to an interior facet because the dual graph is connected and has more than one vertex. The local space $W_K$ therefore contains a basis flux with nonzero total outward flux, so $D_K:W_K\to \mathbb{R}$, $D_{K}v={\int }_K\mathrm{div}v$, is onto. The homogeneous local saddle block has equations $a_K(u,v)-p{D}_{K}v=0$ for all $v\in W_K$ and $D_{K}u=0$. Taking $v=u$ gives $a_K(u,u)=0$, so [F6] gives $u=0$; then surjectivity of $D_K$ forces $p=0$. The square finite-dimensional block is injective and hence invertible.

Let $c=\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\sum }_K\mathrm{\mid }K\mathrm{\mid }{p}_K$ and set ${\bar{p}}_K=p_K-c$. Then $\bar{p}\in Q_h$. Sum the local first equations against a conforming test $v_h\in V_h$. The facet-multiplier terms cancel by normal continuity; the constant pressure term contributes $-c{\int }_D\mathrm{div}{v}_h=0$ by [F7]. The remaining equation is $a(u_h,v_h)-{\int }_D\bar{p}\mathrm{div}{v}_h=0$. Thus $(u_h,\bar{p})$ is the conforming mixed solution, which is unique by [F3].

The map $J$ is onto: for each interior facet $F$, use [F5] on one adjacent cell with zero flux on its other facets and zero field on all other cells; its image is the coordinate vector for $F$. Since $\mathrm{\ell }$ vanishes on $\ker J$, the formula $\mu (Jv):=\mathrm{\ell }(v)$ is well-defined and linear on ${\mathbb{R}}^{{\mathcal{F}}_h^i}$. Write $\mu (y)={\sum }_F{\mu }_F{y}_F$ and choose ${\lambda }_F=-{\mu }_F$. Then $\mathrm{\ell }(v)+{\sum }_F{\lambda }_F{J}_F(v)=0$ for all broken $v$. Taking tests supported in one cell yields each local hybrid equation; balances were already shown, so $(u_h,p_h,\lambda )$ is hybrid. This proves the correspondence before quotienting the pressure gauge.

Invert the local blocks to express each $(u_K,p_K)$ as an affine function of the multiplier vector $\lambda$, with source-dependent right sides. Substitution into $J_F(u)=0$ gives a linear system solely on the finitely many interior-facet multipliers; this is static condensation. Existence and correspondence follow from steps 1.1–2.2. For its homogeneous nullspace, any multiplier vector yields a homogeneous hybrid solution, hence by step 2.1 a conforming solution with zero source. Uniqueness [F3] forces $u=0$ and $p_K=c$ for one global constant. The local equations then read ${\sum }_{F\subset \mathrm{\partial }K,F\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}({\lambda }_F-c){\int }_F{v}_K\cdot n_K=0$ for all $v_K\in W_K$, since Gauss–Green and the zero physical-boundary flux give $D_K{v}_K={\sum }_{F\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{\int }_F{v}_K\cdot n_K$. Using the independent facet basis [F5] gives ${\lambda }_F=c$ on every interior facet. Conversely a common constant vector is a null vector by the gauge identity in [F1]. Thus the kernel is exactly the constant gauge.

For one cell there are no interior multipliers. All its facets are physical boundary facets, so zero boundary flux and local unisolvence [F2] force $W_K=\{0\}$; also $Q_h=\{0\}$. The hybrid cell pressure is arbitrary, but every two such values differ by the common gauge, so the quotient contains one solution, corresponding to the zero conforming pair.