Shape-regular surface meshes and conforming P0 densities

Let $\mathrm{\Gamma }$ be a polyhedral boundary as in Lipschitz polyhedral boundary, orientation, and trace spaces. A conforming surface triangulation ${\mathcal{T}}_h$ is a finite collection of nondegenerate closed planar triangles covering $\mathrm{\Gamma }$, with disjoint relative interiors, such that two distinct triangles meet only in a common full edge, a common vertex, or not at all. For $T\in {\mathcal{T}}_h$, write $h_T=\mathrm{diam}(T)$ and let ${\rho }_T$ be the radius of the largest disk contained in $T$; the family is shape regular if ${\sup }_{h,T}{h}_T\mathrm{/}{\rho }_T<\mathrm{\infty }$, and quasi-uniform if ${\inf }_{h,T}{h}_T\mathrm{/}h>0$ for $h={\max }_{T\in {\mathcal{T}}_h}{h}_T$.

The conforming P0 density space is $X_h=\{{\lambda }_h\in L^2(\mathrm{\Gamma }):{\lambda }_h{\mathrm{\mid }}_T\text{ is constant for every }T\in {\mathcal{T}}_h\}$. The continuous inclusion $H^{1\mathrm{/}2}(\mathrm{\Gamma })\hookrightarrow L^2(\mathrm{\Gamma })$ follows because each chart Bessel norm controls its L2 norm and the fixed partition is finite. Hence every ${\lambda }_h$ defines a bounded functional $v\mapsto {\int }_{\mathrm{\Gamma }}{\lambda }_{h}v$ on $H^{1\mathrm{/}2}(\mathrm{\Gamma })$; thus $X_h\subset H^{-1\mathrm{/}2}(\mathrm{\Gamma })$. This inclusion makes the single-layer Galerkin method conforming in its energy space.