A local moment interpolant for conforming P1 meshes

Assume Countable Choice for the $H_0^1$ convention of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Fix a finite conforming triangular mesh $\mathcal{T}$. For each interior vertex $z$, choose the incident triangle with least fixed mesh index and denote it $K_z$. On the reference triangle $\widehat{K}$, let ${\widehat{\lambda }}_1,{\widehat{\lambda }}_2,{\widehat{\lambda }}_3$ be its barycentric coordinates. The Gram matrix $G_{ij}={\int }_{\widehat{K}}{\widehat{\lambda }}_i{\widehat{\lambda }}_j$ is positive definite because ${\int }_{\widehat{K}}{p}^2>0$ for every nonzero $p\in P_1(\widehat{K})$. Thus there are unique dual affine polynomials ${\widehat{\psi }}_i\in P_1(\widehat{K})$ with ${\int }_{\widehat{K}}{\widehat{\lambda }}_j{\widehat{\psi }}_i={\delta }_{ij}$. Pull them to $K_z$ by the affine map using the normalized measure so ${\int }_{K_z}{\phi }_y{\psi }_{z,K_z}={\delta }_{zy}$ for its three local nodal hats.

For $v\in H_0^1(\mathrm{\Omega })$ set $c_z(v)={\int }_{K_z}v{\psi }_{z,K_z}$ at interior vertices and $c_z(v)=0$ at boundary vertices. Define $I_{\mathcal{T}}v={\sum }_{z\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}{c}_z(v){\phi }_z,$ where ${\phi }_z$ is the continuous piecewise-affine nodal hat. The sum is finite and belongs to the conforming zero-boundary P1 space. This is a moment quasi-interpolant; it is not a point-evaluation operator on arbitrary Sobolev classes.

The Sobolev spaces and zero-boundary closure use Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure. The mesh and finite-element conventions use Matching affine simplicial meshes and shape regularity and Broken and conforming polynomial spaces on a matching mesh. Reproduction and bounds are established by a separate theorem on this page.