Averaged pullback Whitney moments and uniform near identity

On a shape-regular cell, change variables to a fixed reference tetrahedron using the affine differential-form pullback. To verify the scaling, let $A_K$ be the linear part of its affine reference map. Its columns are edge vectors, so $\mathrm{\parallel }{A}_K\mathrm{\parallel }\le \sqrt{3}{h}_K$. Shape regularity gives ${\rho }_K\ge h_K\mathrm{/}\sigma$, and the inscribed ball yields $\mathrm{\mid }\det A_K\mathrm{\mid }=6\mathrm{\mid }K\mathrm{\mid }\ge 8\pi (h_K\mathrm{/}\sigma {)}^3$. If $s_1\ge s_2\ge s_3$ are its singular values, then $s_1,s_2\le \sqrt{3}{h}_K$ and $s_1{s}_2{s}_3=\mathrm{\mid }\det A_K\mathrm{\mid }$, so $s_3\ge c_{\sigma }{h}_K$. Thus $s_{\min }(A_K)$ and $\mathrm{\parallel }{A}_K\mathrm{\parallel }$ are both comparable to $h_K$. Reference finite-dimensional norm equivalence now gives $\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{L^2(K)}^2\simeq {\sum }_{\sigma \subset K,\dim \sigma =k}{h}_K^{3-2k}{\mid {\int }_{\sigma }{v}_h\mid }^2,\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}+h_K\mathrm{\parallel }d{v}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(K)}\le C{h}_K^{-3\mathrm{/}2}\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{L^2(K)}\mathrm{.}$ Norm equivalence follows directly because the moment map is an injective linear map on the finite reference space [F2]; the maximum and minimum of its norm on the Euclidean unit sphere are finite and strictly positive. The inverse estimate follows the same way for polynomial coefficients and their first derivatives. These arguments apply to degrees zero through three, taking $d=0$ in top degree.

We first justify all boundary and weak-differential operations used below. Using the smooth field $b$ from [F1], replace $g_h$ temporarily by the constant one and let $S_t^{+}$ be the average pullback by $x\mapsto x+t(b(x)+ry)$. Its image on $\bar{\mathrm{\Omega }}$ has interior clearance $ct$. In coordinates the pullback coefficients are the minors displayed above, with $D_{x}F=I+tDb(x)$. The change of variables $z=x+t(b(x)+ry)$ writes this operator as integration against a smooth kernel in $x,z$, including the smooth pullback coefficients; hence it maps $L^2$ forms to forms smooth on a neighborhood of $\bar{\mathrm{\Omega }}$. More explicitly, with ${\psi }_t(x)=x+tb(x)$ it is the pullback ${\psi }_t^{\ast }({\check{\rho }}_{tr}\ast \tilde{u})$, where $\check{\rho }(y)=\rho (-y)$ and $\tilde{u}$ is zero extension of the coefficient vector: the pullback minors are independent of $y$. Smoothness thus follows directly from the coefficient convolution theorem [F5] and finite smooth composition, rather than from an unstated parameter differentiation theorem. Pullback commutes with weak $d$: locally mollify on the interior image, apply the smooth pullback identity [F3] there, and pass in $L^2$ using [F5] for the finite exterior-derivative sums and the change of variables [F6]. Averaging preserves this identity. The same change of variables bounds $S_t^{+}$ on $L^2$ uniformly for small $t$. On restrictions of globally smooth forms it tends to the identity, and these restrictions are $L^2$ dense by [F4] (a compactly supported smooth form on the open domain extends smoothly by zero). A three-term approximation then proves $S_t^{+}u\to u$ in $L^2$ for every $u$; applied to $du$ it proves graph convergence for $u\in H{\mathrm{\Lambda }}^k$. The graph differential is closed in the complete finite-product $L^2$ space [F4]: if $u_j\to u$ and $d{u}_j\to f$ in $L^2$, pass each coefficient weak test identity to the limit by Cauchy–Schwarz to obtain $du=f$. In particular smooth-neighborhood forms are graph dense.

Fix $x\in \sigma \subset K$. The map $y\mapsto F_{ϵ,y}^{\pm }(x)$ is translation and scalar dilation with Jacobian $(ϵr{g}_h(x){)}^3$. The coefficients of its pullback on the tangent space of $\sigma$ are bounded by [F1]. Cauchy–Schwarz [F4] in $y$, $g_h\simeq h_K$, and [F1] therefore give ${\int }_{B_1}\rho (y)\mathrm{\mid }((F_{ϵ,y}^{\pm }{)}^{\ast }u{)}_x\mathrm{\mid }dy\le C{ϵ}^{-3\mathrm{/}2}{h}_K^{-3\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\omega }_K)}\mathrm{.}$ For the minus sign use the zero extension; its sampled $L^2$ norm is bounded by the same patch norm. Integrating over $\sigma$, whose $k$-volume is at most $C{h}_K^k$, bounds each moment by $C{ϵ}^{-3\mathrm{/}2}{h}_K^{k-3\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2({\omega }_K)}$. This also proves independence of representatives and absolute integrability. Reconstruction and step 1.1 give the first claimed bound after summing the bounded-overlap patches. At boundary simplices the outward images lie strictly outside $\bar{\mathrm{\Omega }}$, so every minus moment there vanishes.

For a piecewise Whitney form with all boundary tangential moments zero, zero extension has distributional $d$ equal to its piecewise $d$, extended by zero. Indeed integrate cellwise against a compact test form: the interior-face terms cancel by matching tangential traces [F2], and boundary terms vanish because zero boundary moments give zero entire tangential polynomial trace. Use $S_t^{-}$, defined as in step 1.2 with $-b$, on that zero extension. It is smooth and zero on a collar of the boundary, since every outward sample from sufficiently near the boundary lies outside $\bar{\mathrm{\Omega }}$. It converges in the graph norm by the same pullback and density argument. Therefore the relative Whitney form belongs to the graph closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, as asserted. For a general member of that closure, zero extension commutes with $d$ by passage from compactly supported approximants. Conversely, if a Whitney form belongs to this closure, its zero extension has no boundary surface contribution to $d$. Cellwise integration by parts identifies that contribution with its polynomial tangential trace on the relative interior of each boundary face. Such a surface distribution cannot be an $L^2$ function unless its coefficient is zero: testing with a fixed smooth tangential function times a normal cutoff of width $\delta$ leaves a fixed surface pairing while its $L^2$ volume pairing tends to zero. Thus the tangential polynomial trace vanishes on every boundary face. Its boundary subsimplex moments vanish as well. This proves the stated equality of spaces; top degree has no boundary moments or trace condition.

For $k=0,1,2$ and smooth forms, Stokes and pullback commutation [F3] on $\sigma$ give $m_{\sigma }^{\pm ,k+1}(du)={\sum }_{\tau \subset \mathrm{\partial }\sigma }\mathrm{sgn}(\tau ,\sigma )m_{\tau }^{\pm ,k}(u)$. The restriction of $g_h$ to a simplex is affine and common to all incident cells; thus its tangential derivative and the shift on every face agree, even though ambient derivatives jump across mesh faces. Stokes is valid for this smooth restriction. Reconstruction [F2] proves commutation for smooth full inputs and compactly supported relative inputs in degrees zero through two. Use step 1.2 for full graph density and the defining closure for relative graph density. The $L^2$ bounds in step 2.1 and closedness of distributional $d$ pass these identities to precisely those graph domains. In degree three, $d$ vanishes on both the input and reconstructed form, so the terminal identity is zero.

To obtain a bound tending to zero with $ϵ$, use the prism homotopy $H_y(s,x)=x+sϵg_h(x)(\pm b(x)+ry)$ on $[0,1]\times \sigma$. Its $s$-derivative has size at most $Cϵh_K$ and its derivatives along $\sigma$ have size at most $C$. Stokes [F3] on the finitely triangulated prism, equivalently the fundamental theorem of calculus for the displayed pullback-minor coefficients, gives ${\int }_{\sigma }(F_{ϵ,y}^{\pm }{)}^{\ast }{v}_h-{\int }_{\sigma }{v}_h={\int }_{[0,1]\times \sigma }{H}_y^{\ast }(d{v}_h)+{\int }_{[0,1]\times \mathrm{\partial }\sigma }{H}_y^{\ast }{v}_h,$ with the standard oriented prism signs; only absolute values will be used. In the relative case take the zero extension from step 2.2. To justify this identity for the piecewise polynomial form, approximate full forms by $S_t^{+}{v}_h$ and relative zero extensions by whole-space convolutions. Their coefficients and exterior derivatives are bounded on the pertinent patches by the piecewise bounds of step 1.1, uniformly as $t\downarrow 0$. On the original simplex their tangential traces converge to the common Whitney tangential trace: for an interior point of a mesh simplex, all sufficiently near cells contain that simplex and have that same trace; the pullback's tangent directions tend to the original ones. On the shifted simplex convergence holds for almost every $y$ and almost every $x$, because for fixed $x$ the map $y\mapsto F_{ϵ,y}^{\pm }(x)$ has nonzero volume Jacobian; change of variables and Fubini [F6] therefore give exactly this null-set passage. The same argument works inside the prism for every $s>0$. Dominated convergence [F6], first in simplex parameters and $y$ and then in $s$, passes the averaged identity and its estimates to $v_h$. At $s=0$ the trace argument just given supplies the original moment.

The first prism integral is bounded by $Cϵh_K^{k+1}\mathrm{\parallel }d{v}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\omega }_K)}$ and the second by $Cϵh_K^k\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\omega }_K)}$. For $k=0$ the second term is absent; for $k=3$ the first is zero. Every cell in the patch has diameter comparable to $h_K$. Step 1.1 therefore bounds their sum by $Cϵh_K^{k-3\mathrm{/}2}\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{L^2({\omega }_K)}$. Average in $y$, reconstruct using step 1.1, and sum the bounded-overlap patches. This gives $\mathrm{\parallel }(Q_h^{\pm ,k}-I)v_h{\mathrm{\parallel }}_2\le Cϵ\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_2$ uniformly in every degree.