A conforming averaging companion for Crouzeix–Raviart functions

Vertex stars are facet-connected. Fix a mesh vertex $z$. Finiteness of the mesh lets us choose a ball $B_R(z)$ meeting no closed cell that does not contain $z$; within this ball the union of incident cells is exactly the polyhedral cone obtained by joining $z$ to their opposite facets. Its intersection $L_z$ with a sufficiently small centered sphere is a finite face-to-face spherical triangulation. At an interior vertex $L_z$ is the whole sphere. At a boundary vertex, the Lipschitz polyhedral definition in [given] gives, after a rigid motion and shrinking $R$, an epigraph $C_z=\{(x^{\mathrm{\prime }},t):t>\psi (x^{\mathrm{\prime }})\}$ where $\psi$ is a positively homogeneous piecewise-linear Lipschitz function; this is the local tangent cone, and radial projection maps $C_z\setminus \{0\}$ onto $L_z$. The cone $C_z$ is path-connected: for two points, join each vertically to a height $H$ above the maximum of $\psi$ on the horizontal segment between their base coordinates, then join horizontally at height $H$. Radial projection therefore makes $L_z$ path-connected. Any two interiors of top-dimensional spherical simplices in this link can be joined by a compact path in $L_z$. In spherical charts covering that path, each codimension-two simplex lies in a set of dimension at most $m-2$, where $m=d-1$; for a fixed path endpoint, next endpoints whose straight chart segment meets such a set lie in a cone of dimension at most $m-1$, hence have empty interior. Choose the perturbation balls small enough to remain in $L_z$ and successively avoid the finitely many such cones. The resulting path avoids the codimension-two skeleton and changes simplices only across codimension-one faces. Hence the cells incident to $z$ form a connected graph using facets containing $z$. At a boundary vertex some boundary mesh facet contains $z$, so the star graph reaches a boundary facet. Its number of cells is uniformly bounded by [F1].

For an interior mesh vertex $z$, define $(E_h{v}_h)(z)$ as the arithmetic mean of the values $v_K(z)$ over cells $K$ incident to $z$. For a boundary vertex set $(E_h{v}_h)(z)=0$. On each cell define $E_h{v}_h$ as the affine polynomial with these vertex values. Adjacent cell restrictions agree at every vertex of their common facet, hence agree on the whole facet, so $E_h{v}_h$ is continuous and piecewise affine. Every boundary facet has all its vertices assigned zero, so its polynomial trace is identically zero.

If cells $K,L$ share a facet $F$ containing a vertex $z$, then the restriction of an affine function $p$ satisfies $p(x)=p(b_F)+{\mathrm{\nabla }}_{F}p\cdot (x-b_F)$ on $F$, and ${\int }_F(x-b_F)dS=0$ by the definition of the barycenter; hence its face mean is $p(b_F)$. By [F2] the means of $v_K,v_L$ agree, and therefore $\mathrm{\mid }{v}_K(z)-v_L(z)\mathrm{\mid }\le h_F(\mathrm{\mid }\mathrm{\nabla }{v}_K\mathrm{\mid }+\mathrm{\mid }\mathrm{\nabla }{v}_L\mathrm{\mid })$. If $F$ is a boundary facet containing $z$, its mean is zero, so $\mathrm{\mid }{v}_K(z)\mathrm{\mid }\le h_F\mathrm{\mid }\mathrm{\nabla }{v}_K\mathrm{\mid }$. Following the bounded facet paths of step 1.1 gives, for every cell $K$ and vertex $z\in K$, $\mathrm{\mid }{v}_K(z)-(E_h{v}_h)(z)\mathrm{\mid }\le C{h}_K{\sum }_{L\ni z}\mathrm{\mid }\mathrm{\nabla }{v}_L\mathrm{\mid }$, where adjacent diameters are comparable by [F1]. The affine barycentric formula on $K$ now gives $\mathrm{\parallel }{v}_h-E_h{v}_h{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K{\sum }_{L\cap K\text{ shares a vertex}}\mathrm{\parallel }\mathrm{\nabla }{v}_L{\mathrm{\parallel }}_{L^2(L)}$ and $\mathrm{\parallel }\mathrm{\nabla }(E_h{v}_h-v_K){\mathrm{\parallel }}_{L^2(K)}\le C{\sum }_{L\cap K\text{ shares a vertex}}\mathrm{\parallel }\mathrm{\nabla }{v}_L{\mathrm{\parallel }}_{L^2(L)}$. Each patch contains a uniformly bounded number of cells and overlaps only a uniformly bounded number of times by [F1]; summing the squared inequalities proves both displayed norm bounds, except for $H_0^1$ membership.

It remains to prove $E_h{v}_h\in H_0^1(\mathrm{\Omega })$. Extend $E_h{v}_h$ by zero to $\tilde{E}$ on ${\mathbb{R}}^d$. Apply [F3] cellwise to $E_h{v}_h\phi e_i$ for any $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$. Interior fluxes cancel because the traces of $E_h{v}_h$ agree, and boundary fluxes vanish because its boundary-facet trace is zero. Thus the zero extension has the cellwise gradient as its weak gradient and belongs to $W^{1,2}({\mathbb{R}}^d)$. Let $M$ be the maximum of the finitely many cellwise gradient magnitudes. Restricting $\tilde{E}$ to any line segment partitions it into finitely many cell/exterior pieces; on each piece it is affine with slope at most $M$, and continuity at piece endpoints gives $\mathrm{\mid }\tilde{E}(x)-\tilde{E}(y)\mathrm{\mid }\le M\mathrm{\mid }x-y\mathrm{\mid }$. If a segment lies in a mesh face, its restriction is affine there with the same bound. Hence $\tilde{E}$ is globally Lipschitz. For $d_{\mathrm{\Omega }}(x)=\mathrm{dist}(x,{\mathbb{R}}^d\setminus \mathrm{\Omega })$, the triangle inequality gives $\mathrm{\mid }{d}_{\mathrm{\Omega }}(x)-d_{\mathrm{\Omega }}(y)\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$; since $\tilde{E}$ vanishes on the complement, $\mathrm{\mid }{E}_h{v}_h(x)\mathrm{\mid }\le M{d}_{\mathrm{\Omega }}(x)$.

Let $\rho$ be the nonnegative unit-mass smooth bump supplied by [F7], let $\theta$ be supplied by [F8], and set $g_{ϵ}(x)=\theta (d_{\mathrm{\Omega }}(x)\mathrm{/}ϵ)$ and ${\eta }_{ϵ}=g_{ϵ}\ast {\rho }_{ϵ\mathrm{/}4}$. The function $g_{ϵ}$ is bounded and $C\mathrm{/}ϵ$-Lipschitz; convolution with the compactly supported smooth bump makes ${\eta }_{ϵ}$ smooth and preserves this Lipschitz bound, so $\mathrm{\parallel }\mathrm{\nabla }{\eta }_{ϵ}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\mathrm{/}ϵ$. Since ${\rho }_{ϵ\mathrm{/}4}$ is supported in the radius-$ϵ\mathrm{/}4$ ball, $d_{\mathrm{\Omega }}(x)<3ϵ\mathrm{/}4$ implies $g_{ϵ}(x-y)=0$ throughout the convolution neighborhood, while $d_{\mathrm{\Omega }}(x)\ge 9ϵ\mathrm{/}4$ implies $g_{ϵ}(x-y)=1$ there. Therefore $\mathrm{supp}{\eta }_{ϵ}\subseteq \{d_{\mathrm{\Omega }}\ge 3ϵ\mathrm{/}4\}$, a closed bounded subset of $\mathrm{\Omega }$ and hence compact by [F11], and $0\le {\eta }_{ϵ}\le 1$ with ${\eta }_{ϵ}=1$ on $\{d_{\mathrm{\Omega }}\ge 9ϵ\mathrm{/}4\}$. The product rule [F6] shows that $\tilde{E}-{\eta }_{ϵ}\tilde{E}$ and its weak gradient are supported in $S_{ϵ}=\{x\in \mathrm{\Omega }:d_{\mathrm{\Omega }}(x)<9ϵ\mathrm{/}4\}$. On this layer $\mathrm{\mid }{E}_h{v}_h\mathrm{\mid }\le CMϵ$ and $\mathrm{\mid }{E}_h{v}_h\mathrm{\nabla }{\eta }_{ϵ}\mathrm{\mid }\le CM$, while $\mathrm{\mid }\mathrm{\nabla }\tilde{E}\mathrm{\mid }\le M$ almost everywhere. Thus $\mathrm{\parallel }\tilde{E}-{\eta }_{ϵ}\tilde{E}{\mathrm{\parallel }}_{W^{1,2}({\mathbb{R}}^d)}^2\le C{M}^2\mathrm{\mid }{S}_{ϵ}\mathrm{\mid }$. The sets $S_{1\mathrm{/}n}$ decrease to the empty set and are contained in the finite-measure set $\mathrm{\Omega }$ by [F9]; continuity from above [F5] gives $\mathrm{\mid }{S}_{1\mathrm{/}n}\mathrm{\mid }\to 0$. For $ϵ\le 1\mathrm{/}n$, $S_{ϵ}\subseteq S_{1\mathrm{/}n}$, so ${\eta }_{ϵ}{E}_h{v}_h\to E_h{v}_h$ in $W^{1,2}(\mathrm{\Omega })$.

Fix $ϵ>0$. The function $w_{ϵ}={\eta }_{ϵ}\tilde{E}$ is a compactly supported $W^{1,2}({\mathbb{R}}^d)$ function whose support $K_{ϵ}$ is a compact subset of $\mathrm{\Omega }$. By [F10], choose ${\chi }_{ϵ}\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ equal to one near $K_{ϵ}$. By [F4], choose ${\phi }_n\in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ converging to $w_{ϵ}$ in $W^{1,2}({\mathbb{R}}^d)$. The weak product rule [F6] makes ${\chi }_{ϵ}{\phi }_n\to {\chi }_{ϵ}{w}_{ϵ}=w_{ϵ}$ in $W^{1,2}$, and each product belongs to $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$. Hence $w_{ϵ}{\mathrm{\mid }}_{\mathrm{\Omega }}\in H_0^1(\mathrm{\Omega })$. By step 4.1, these functions converge to $E_h{v}_h$ in $W^{1,2}(\mathrm{\Omega })$ as $ϵ\downarrow 0$; the closure definition [F4] proves $E_h{v}_h\in H_0^1(\mathrm{\Omega })$.