Boundary-preserving quasi-interpolation is stable and strongly approximating

The coefficients define a linear map into the algebraic conforming piecewise-polynomial space: shared-face nodes coincide, so the face polynomials have identical nodal data and hence identical traces; all boundary coefficients are zero, so each boundary-facet polynomial vanishes. For $d\ge 2$, [F3] therefore gives $V_h\subset W_0^{1,p}(\mathrm{\Omega })$. For $d=1$, write a mesh function as $w$, let $M$ bound its finitely many piecewise derivatives, and extend it by zero. It is continuous at every mesh node and at every component endpoint. Integrating the derivative on each of the finitely many subintervals between two points using the fundamental theorem in [F6] shows $\mathrm{\mid }w(x)-w(y)\mathrm{\mid }\le M\mathrm{\mid }x-y\mathrm{\mid }$ on $\mathbb{R}$; integration by parts on those subintervals shows that the zero extension has the piecewise derivative as its weak derivative. Thus it is compactly supported in $W^{1,p}(\mathbb{R})$.

Let $u=E_{0}v$. For a cell $K$, choose a cube $C_K$ of side $Rh$ centered at an interior point of $K$, where the fixed $R$ is large enough that $K$, every radius-$h$ ball centered at one of its nodes, and every radius-$2h$ ball centered at a point of $K$ lie in $C_K$. The cubes have bounded overlap: if $x\in C_K$, then $K\subset B(x,C_{R}h)$; the cells have disjoint interiors and volume at least $c{h}^d$, so only uniformly many can meet that ball. For smooth $u$ on a cube $C$, the segment fundamental theorem, Hölder, Jensen's inequality, and a change of variables on the two ranges $t\le 1\mathrm{/}2$ and $t\ge 1\mathrm{/}2$ give $\mathrm{\parallel }u-u_C{\mathrm{\parallel }}_{L^p(C)}\le C_d\mathrm{diam}(C)\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(C)},u_C=\mathrm{\mid }C{\mathrm{\mid }}^{-1}{\int }_{C}u\mathrm{.}$ Indeed, Jensen's inequality and $u(x)-u(y)={\int }_0^1\mathrm{\nabla }u(tx+(1-t)y)\cdot (x-y)dt$ give $\mathrm{\parallel }u-u_C{\mathrm{\parallel }}_{L^p(C)}^p\le \frac{\mathrm{diam}(C{)}^p}{\mathrm{\mid }C\mathrm{\mid }}{\int }_C{\int }_C{\int }_0^1\mathrm{\mid }\mathrm{\nabla }u(tx+(1-t)y){\mathrm{\mid }}^{p}dtdydx\mathrm{.}$ For $t\ge 1\mathrm{/}2$ change the $x$ variable, and for $t\le 1\mathrm{/}2$ change the $y$ variable; both inverse Jacobians are at most $2^d$ and the images stay in $C$. This proves the displayed Poincaré estimate. Since $u=E_{0}v$ is compactly supported in $W^{1,p}({\mathbb{R}}^d)$, [F6] approximates it by smooth functions in that norm, so the same cube estimate holds for $u$.

In the one-dimensional branch, for $x$ in a component interval choose the nearer endpoint; integrating the derivative from that endpoint by [F6] gives $\mathrm{\mid }w(x)\mathrm{\mid }\le M\mathrm{dist}(x,\mathbb{R}\setminus \mathrm{\Omega })$. Because there are finitely many distinct endpoints, choose $\epsilon$ smaller than one quarter of their minimum separation. Translate and rescale the bump from [F9] at each endpoint to obtain ${\chi }_{a,\epsilon }$ equal to one within $\epsilon$ of $a$ and zero outside distance $2\epsilon$. The endpoint supports are disjoint; the product ${\eta }_{\epsilon }={\prod }_{a\in \mathrm{\partial }\mathrm{\Omega }}(1-{\chi }_{a,\epsilon })$ is smooth, vanishes within distance $\epsilon$ of each endpoint, equals one at distance at least $2\epsilon$, and satisfies $\mathrm{\mid }{\eta }_{\epsilon }^{\mathrm{\prime }}\mathrm{\mid }\le C\mathrm{/}\epsilon$. Thus $w-{\eta }_{\epsilon }w$ and its weak derivative are supported in a boundary layer of measure $O_{\mathrm{\Omega }}(\epsilon )$ and bounded there by $C_{\mathrm{\Omega }}M$. On each partition interval ${\eta }_{\epsilon }w$ is smooth times polynomial, is continuous at nodes, and vanishes near component endpoints; integration by parts on the finitely many intervals cancels all interior endpoint terms. Its zero extension therefore belongs to $W^{1,p}(\mathbb{R})$ with derivative ${\eta }_{\epsilon }^{\mathrm{\prime }}w+{\eta }_{\epsilon }{w}^{\mathrm{\prime }}$ on the pieces. Hence ${\eta }_{\epsilon }w\to w$ in $W^{1,p}(\mathbb{R})$. Each cutoff product has compact support a positive distance from $\mathbb{R}\setminus \mathrm{\Omega }$; convolving it with a sufficiently small mollifier gives a function in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ by [F4] and [F7], and the convolution converges in $W^{1,p}$ by [F4]. This proves $w\in W_0^{1,p}(\mathrm{\Omega })$ by [F8] and completes the range assertion in $d=1$.

If $K$ has no boundary node, put $c_K=u_{C_K}$. If it has a boundary node, put $c_K=0$. In $d\ge 2$, compactness of the boundary gives a finite subcover of its Lipschitz graph charts; taking the largest chart Lipschitz constant and the smallest chart radius yields uniform $\eta ,r_0>0$ such that each boundary point $z$ and $0<r\le r_0$ have an exterior cone sector in $B(z,r)$ of measure at least $\eta r^d$: in a chart $x_d>g(x^{\mathrm{\prime }})$ with Lipschitz constant $L$, the cone $x_d<g(z^{\mathrm{\prime }})-L\mathrm{\mid }{x}^{\mathrm{\prime }}-z^{\mathrm{\prime }}\mathrm{\mid }$ is exterior, and a fixed smaller cone has that volume. For sufficiently small $h$, a boundary node $z$ of $K$ therefore gives $\mathrm{\mid }{C}_K\setminus \mathrm{\Omega }\mathrm{\mid }\ge {\eta }^{\mathrm{\prime }}\mathrm{\mid }{C}_K\mathrm{\mid }$. In $d=1$, at a boundary endpoint the exterior interval inside $C_K$ has a fixed positive fraction of its length; for finitely many separated components take $h$ smaller than their minimum gap divided by $2R$. In the boundary case $u=0$ on $C_K\setminus \mathrm{\Omega }$, so $\mathrm{\mid }{u}_{C_K}\mathrm{\mid }\mathrm{\mid }{C}_K\setminus \mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}p}\le \mathrm{\parallel }u-u_{C_K}{\mathrm{\parallel }}_{L^p(C_K\setminus \mathrm{\Omega })}\mathrm{.}$ The cube estimate of step 1.2 then gives, in either case, $\mathrm{\parallel }u-c_K{\mathrm{\parallel }}_{L^p(C_K)}\le Ch\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(C_K)}\mathrm{.}$

Fix $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and let $\delta =\mathrm{dist}(\mathrm{supp}\psi ,\mathrm{\partial }\mathrm{\Omega })>0$. For $h<\delta \mathrm{/}2$, $S_h\psi$ is zero on every cell with a boundary node: every point of such a cell is within distance $h$ of the boundary, while the convolution samples only a radius-$h$ ball. Thus $Q_h\psi =0=S_h\psi$ on those cells. Every other cell has only interior nodes, so $Q_h\psi$ is the ordinary nodal interpolant of $S_h\psi$ there. Since $k\ge 1$, this interpolant reproduces affine polynomials. Taylor's formula about a cell vertex and the local coefficient estimates in [F1] give $\mathrm{\parallel }{Q}_h\psi -S_h\psi {\mathrm{\parallel }}_{L^p(K)}\le C{h}^2\mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}p}\mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{\parallel }\mathrm{\nabla }(Q_h\psi -S_h\psi ){\mathrm{\parallel }}_{L^p(K)}\le Ch\mathrm{\mid }K{\mathrm{\mid }}^{1\mathrm{/}p}\mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$ The nonnegative unit-mass kernel gives $S_h\psi \to \psi$ in $W^{1,p}(\mathrm{\Omega })$ and $\mathrm{\parallel }{D}^2{S}_h\psi {\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{D}^2\psi {\mathrm{\parallel }}_{\mathrm{\infty }}$. Summing cell bounds proves $Q_h\psi \to \psi$ in $W^{1,p}$.

For each interior node $z$ of $K$, Hölder [F5] gives $\mathrm{\mid }{S}_{h}v(z)-c_K\mathrm{\mid }\le \mathrm{\parallel }{\rho }_h{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}({\mathbb{R}}^d)}\mathrm{\parallel }u-c_K{\mathrm{\parallel }}_{L^p(B(z,h))}\le C{h}^{-d\mathrm{/}p}\mathrm{\parallel }u-c_K{\mathrm{\parallel }}_{L^p(C_K)}\mathrm{.}$ At a boundary node both the coefficient and $c_K$ are zero in the boundary case. If $K$ has no boundary node, every local node is interior and the local nodal basis reproduces the constant $c_K$. Applying the local coefficient estimate of [F1] to the coefficients of $Q_{h}v-c_K$, and using step 2.2, gives $\mathrm{\parallel }{Q}_{h}v-c_K{\mathrm{\parallel }}_{L^p(K)}\le Ch\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(C_K)},\mathrm{\parallel }\mathrm{\nabla }{Q}_{h}v{\mathrm{\parallel }}_{L^p(K)}\le C\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(C_K)}\mathrm{.}$ The boundary-node case uses $c_K=0$; the no-boundary-node case uses ${\sum }_{z\in K}{\varphi }_z=1$.

Since $v=u$ on $K$, step 2.2 and the first estimate of step 3.1 imply $\mathrm{\parallel }v-Q_{h}v{\mathrm{\parallel }}_{L^p(K)}\le Ch\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^p(C_K)}\mathrm{.}$ Raise this and the gradient estimate of step 3.1 to the $p$th power, sum over cells, and use the bounded overlap from step 1.2 and the norm preservation in [F2]. This proves both global estimates in the Statement. Also $\mathrm{\parallel }{Q}_{h}w{\mathrm{\parallel }}_{W^{1,p}}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{1,p}}$ for every $w\in W_0^{1,p}$ and all sufficiently small $h$, by the gradient bound and $\mathrm{\parallel }{Q}_{h}w{\mathrm{\parallel }}_p\le \mathrm{\parallel }w{\mathrm{\parallel }}_p+Ch\mathrm{\parallel }\mathrm{\nabla }w{\mathrm{\parallel }}_p$.

For arbitrary $v\in W_0^{1,p}(\mathrm{\Omega })$ and $\epsilon >0$, [F8] provides $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\mathrm{\parallel }v-\psi {\mathrm{\parallel }}_{W^{1,p}}<\epsilon$. The uniform bound in step 4.1 and the smooth convergence in step 2.3 imply ${\mathrm{lim\hspace{0.17em}sup}}_{h\to 0}\mathrm{\parallel }{Q}_{h}v-v{\mathrm{\parallel }}_{W^{1,p}}\le (C+1)\mathrm{\parallel }v-\psi {\mathrm{\parallel }}_{W^{1,p}}<(C+1)\epsilon \mathrm{.}$ Letting $\epsilon \downarrow 0$ proves strong convergence for every $v\in W_0^{1,p}(\mathrm{\Omega })$.