Boundary-preserving face-average P1 interpolation

On each reference facet, the P1 facet mass matrix is positive definite because ${\int }_F{p}^2>0$ for every nonzero affine polynomial. Its inverse defines the dual functions ${\widehat{\psi }}_z$; affine scaling in [F2] gives $\mathrm{\parallel }{\psi }_{z,F}{\mathrm{\parallel }}_{L^2(F)}\le C{h}_F^{-(d-1)\mathrm{/}2}$. The trace map in [F3] therefore makes every coefficient of $I_{h}v$ a bounded linear functional on the H1 trace of an adjacent cell. Shared coefficients define a continuous P1 function. If $v\in H_0^1$, [F4] makes all selected boundary-facet averages zero, so $I_{h}v$ has zero boundary nodal values and belongs to $H_0^1$ by [F1].

The coefficients reproduce every global affine polynomial: on the selected facet its trace is P1 and the dual moment equals its value at $z$. Shape regularity bounds the number of cells at each vertex: each simplex's vertex solid angle has a positive lower bound, while the total angle is at most $2\pi$ in dimension two or $4\pi$ in dimension three. The vertex-star graph is connected as follows. Choose a small chart cylinder meeting only cells incident to the vertex. At an interior vertex, a small ball lies in $D$. At a boundary vertex, the Lipschitz chart represents this local part of $D$ as the region above a Lipschitz graph; within a smaller base ball, vertical paths to a common height above the graph and a horizontal segment at that height connect any two points. Hence this local part is path connected. Join interior points of any two incident cells by a polygonal path there and perturb its finitely many segments to avoid the finite mesh skeleton of codimension at least two. The path then crosses only facet interiors and gives a facet-adjacency chain among cells incident to the vertex. Adjacent cell diameters are comparable by [F2]. Thus vertex-patch diameters and overlap are uniformly bounded.

Let ${\omega }_K$ be the union of cells meeting a vertex of $K$. Shape regularity bounds the number of cells at each vertex: each has a solid angle bounded below by the reference-cell shape bound, whereas the total angle is at most $2\pi$ in dimension two and $4\pi$ in dimension three. Each vertex-star graph is connected by step 1.2, and all stars of vertices of $K$ contain $K$, so their union is connected; neighboring cell diameters are comparable by [F2]. For adjacent cells $T,T^{\mathrm{\prime }}$ with common facet $F$, let $m_T,m_{T^{\mathrm{\prime }}}$ be their volume means. Cell Poincare and trace in [F3] give $\mathrm{\parallel }w-m_T{\mathrm{\parallel }}_{L^2(F)}\le C{h}_K^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(T)}$ and the analogous bound on $T^{\mathrm{\prime }}$. Since $\mathrm{\mid }F{\mathrm{\mid }}^{1\mathrm{/}2}\asymp h_K^{(d-1)\mathrm{/}2}$, it follows that $\mathrm{\mid }{m}_T-m_{T^{\mathrm{\prime }}}\mathrm{\mid }\le C{h}_K^{(2-d)\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2(T\cup T^{\mathrm{\prime }})}$. A path of bounded length through the patch graph then compares all cell means to one fixed $c_K$ and proves $\mathrm{\parallel }w-c_K{\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Applying the scaled trace estimate of [F3] and $\mathrm{\parallel }{\psi }_{z,F_z}{\mathrm{\parallel }}_{L^2(F_z)}\le C{h}_K^{-(d-1)\mathrm{/}2}$ gives $\mathrm{\mid }{\int }_{F_z}{\psi }_{z,F_z}(w-c_K)\mathrm{\mid }\le C{h}_K^{1-d\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$ for every vertex of $K$.

Suppose $v\in H^2(D)$. On each cell $T\subset {\omega }_K$, Bramble--Hilbert and affine scaling in [F2, F3] give $p_T\in P_1(T)$ with $\mathrm{\parallel }v-p_T{\mathrm{\parallel }}_{L^2(T)}\le C{h}_T^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2(T)}$ and $\mathrm{\parallel }\mathrm{\nabla }(v-p_T){\mathrm{\parallel }}_{L^2(T)}\le C{h}_T\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2(T)}$. For adjacent cells $T,T\mathrm{\prime }$ sharing a facet $F$, the scaled trace estimate applied to $v-p_T$ and its first derivatives gives $\mathrm{\parallel }{p}_T-p_{T\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(F)}\le C{h}_K^{3\mathrm{/}2}\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2(T\cup T\mathrm{\prime })}$ and $\mathrm{\mid }F{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\mid }\mathrm{\nabla }{p}_T-\mathrm{\nabla }{p}_{T\mathrm{\prime }}\mathrm{\mid }\le C{h}_K^{1\mathrm{/}2}\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2(T\cup T\mathrm{\prime })}$. The second bound controls the full gradient difference, and the first then controls its constant part. Normalize both cell diameters by $h_K$. Shape regularity and comparability across the shared facet bound the normalized affine maps and their inverses; there are only finitely many vertex-orderings. Their closure is a compact set of nondegenerate reference-pair configurations. Finite-dimensional norm equivalence [F5] varies continuously on this compact set, so its constants are uniform; affine scaling then gives $\mathrm{\parallel }{p}_T-p_{T\mathrm{\prime }}{\mathrm{\parallel }}_{H^1(T)}\le C{h}_K\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2(T\cup T\mathrm{\prime })}$. A bounded path through the connected cell-star graph lets one fixed cell polynomial $p\in P_1$ satisfy $\mathrm{\parallel }v-p{\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K^2\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$ and $\mathrm{\parallel }\mathrm{\nabla }(v-p){\mathrm{\parallel }}_{L^2({\omega }_K)}\le C{h}_K\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. This uses only local approximation and does not impose boundary values on $v$.

The reference P1 basis satisfies $\mathrm{\parallel }{\phi }_z{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{d\mathrm{/}2}$ and $\mathrm{\parallel }\mathrm{\nabla }{\phi }_z{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K^{d\mathrm{/}2-1}$. Since $I_h$ reproduces constants, step 2.1 bounds $\mathrm{\parallel }{I}_{h}w-c_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$ and $\mathrm{\parallel }\mathrm{\nabla }{I}_{h}w{\mathrm{\parallel }}_{L^2(K)}\le C\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Together with the patch Poincare bound for $w-c_K$, these prove the $L^2$ approximation and local H1 stability. Summing and using bounded overlap proves global H1-seminorm stability.

Facet duality reproduces every affine polynomial at every vertex, so $I_{h}p=p$ on $K$. Apply the local H1 stability from step 3.1 to $v-p$ and use step 2.2 to get $\mathrm{\parallel }\mathrm{\nabla }(v-I_{h}v){\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }{D}^{2}v{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Summing over cells and using the bounded patch overlap from step 3.1 proves the global H1 approximation estimate.