Uniform Fortin stability of the mini element

For each cell put $c_K=({\int }_K{b}_K{)}^{-1}{\int }_K(v-I_{h}v)\in {\mathbb{R}}^d$, and define ${\mathrm{\Pi }}_{h}v=I_{h}v+{\sum }_K{b}_K{c}_K$. The denominator is positive because $b_K>0$ in the cell interior. The added functions vanish on cell boundaries, so ${\mathrm{\Pi }}_{h}v\in V_h$.

We track the constants in the construction of [F1]. Shape regularity bounds each vertex solid angle below, so at most $N(d,\sigma )$ cells meet a vertex, where $\sigma$ is the shape-regularity bound. Each vertex star is facet connected: a sufficiently small interior ball or boundary Lipschitz-epigraph neighborhood is path connected, and paths avoiding the codimension-two mesh skeleton cross only facets. Thus cells in ${\omega }_K$ are linked by uniformly bounded facet chains and have comparable diameters by [F6]. For adjacent cells $T,T^{\mathrm{\prime }}$ with means $m_T,m_{T^{\mathrm{\prime }}}$, cell Poincare and trace give $\mathrm{\mid }{m}_T-m_{T^{\mathrm{\prime }}}\mathrm{\mid }\le C{h}_K^{1-d\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(T\cup T^{\mathrm{\prime }})}$. Summing along those chains and choosing $c$ to be one cell mean gives $\mathrm{\parallel }v-c{\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. The facet dual functions of [F1] have norm at most $C{h}_K^{-(d-1)\mathrm{/}2}$; the scaled trace inequality therefore gives $\mathrm{\mid }{\int }_{F_z}{\psi }_{z,F_z}(v-c)\mathrm{\mid }\le C{h}_K^{1-d\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}$ for each vertex of $K$. Since $I_h$ reproduces constants, the local P1 basis bounds yield $\mathrm{\parallel }\mathrm{\nabla }{I}_{h}v{\mathrm{\parallel }}_{L^2(K)}+h_K^{-1}\mathrm{\parallel }v-I_{h}v{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. All these constants and the patch-overlap bound depend only on $d,\sigma$; the domain charts were used only for connectivity.

Put $w={\mathrm{\Pi }}_{h}v-v\in H_0^1(D;{\mathbb{R}}^d)$. By [F5], $q_h\in H^1(D)$; approximating $w$ in $H^1$ by compactly supported smooth fields and using the weak-derivative definition of $q_h$ gives $b(w,q_h)={\int }_{D}w\cdot \mathrm{\nabla }{q}_h$. Since $\mathrm{\nabla }{q}_h$ is constant on each cell and step 1.1 gives ${\int }_{K}w=0$, this equals ${\sum }_K\mathrm{\nabla }(q_h{\mathrm{\mid }}_K)\cdot {\int }_{K}w=0$. Therefore $b({\mathrm{\Pi }}_{h}v,q_h)=b(v,q_h)$.

Reference scaling in [F6] gives ${\int }_K{b}_K\ge c_d\mathrm{\mid }K\mathrm{\mid }$ and $\mathrm{\parallel }\mathrm{\nabla }{b}_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{d\mathrm{/}2-1}$. Thus $\mathrm{\parallel }\mathrm{\nabla }(b_K{c}_K){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{-1}\mathrm{\parallel }v-I_{h}v{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_K)}$ by step 1.2. Squaring, summing cells and using bounded patch overlap gives a uniform bound for the bubble correction; adding the H1 stability of $I_h$ from step 1.2 proves $\mathrm{\parallel }{\mathrm{\Pi }}_{h}v{\mathrm{\parallel }}_V\le C_F\mathrm{\parallel }v{\mathrm{\parallel }}_V$.

The construction is linear in $v$ and has the preservation and norm properties in steps 2.1 and 2.2, so [F3] applies with $\beta \ge C_{\mathrm{d}\mathrm{i}\mathrm{v}}(D{)}^{-1}$. It yields ${\beta }_h\ge (C_F{C}_{\mathrm{d}\mathrm{i}\mathrm{v}}(D){)}^{-1}$ uniformly over the mesh family.