Boundary-preserving NURBS projection and matching-patch energy approximation

On $H=H_0^1(D)$ use its full H1 inner product from [F1] as the coercive form $a_0$, and use the functional $v\mapsto (u,v{)}_{H^1}$. By [F12], this functional is bounded by $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$; the form $a_0$ has boundedness and coercivity constants both one. Under [A1], [F2] applies; its continuous solution is $u$ itself and its unique discrete solution is precisely the specified ${\mathrm{\Pi }}_{h}u$. The $M\mathrm{/}\alpha =1$ estimate proves the asserted best-approximation bound. The variational equations and uniqueness imply real linearity. If $u\in V_h$, $u$ satisfies the discrete equations, so ${\mathrm{\Pi }}_{h}u=u$; this proves both surjectivity onto $V_h$ and idempotence, including the zero-space case. Testing with $v_h={\mathrm{\Pi }}_{h}u$ and [F12] gives $\mathrm{\parallel }{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{H^1}^2\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{H^1}$, hence contraction after division when nonzero. All images lie in $V_h\subset H_0^1(D)$.

Put $s_a={\mathcal{S}}_a{g}_a$. By [F3], it belongs to the actual numerator space, is continuous and cellwise polynomial, and vanishes on exterior faces. On a shared face its two traces obey $s_a=c_{ab}{s}_b\circ T_{ab}$, because [F3] restricts to the identical tangential construction and respects multiplication by $c_{ab}$. Dividing by the stated denominator relation gives equality of the rational traces. Therefore the piecewise physical $v_h$ has matching values everywhere and continuous value zero on $\mathrm{\partial }D$; positivity of the denominator and admissibility of the finite maps make it continuous on $\bar{D}$, cellwise C1, and give square-integrable gradients. By [F5], $v_h\in H^1(D)$.

We verify the coordinate-line part of the zero-boundary closure without invoking a C1-domain trace theorem on a Lipschitz domain. Let $z\in H^1(D)$ be real, continuous on $\bar{D}$, cellwise $C^1$ on the finite partition, and zero on $\mathrm{\partial }D$. In one open cell, fix a coordinate $j$ and cover the cell by all open rational boxes whose closures lie in the cell; [F10] makes this a countable cover. On any such box, for each fixed choice of the other coordinates the function of $x_j$ is differentiable. If $z=0$ and ${\mathrm{\partial }}_{j}z\mathrm{\ne }0$ at $x_j=t_0$, the derivative definition gives $\mathrm{\mid }z(t)-z(t_0)\mathrm{\mid }\ge \frac{1}{2}\mathrm{\mid }{\mathrm{\partial }}_{j}z(t_0)\mathrm{\mid }\cdot \mathrm{\mid }t-t_0\mathrm{\mid }$ for all sufficiently close $t\mathrm{\ne }{t}_0$, so zero is isolated among the zeros on that line. Fix an injection of rational intervals into $\mathbb{N}$ using [F10]; assigning to each such zero the interval with least code that contains it and no other zero gives an injection into a countable family. Therefore every coordinate-line section of $Z_j:=\{z=0,{\mathrm{\partial }}_{j}z\mathrm{\ne }0\}$ is null. The set is measurable on the box because $z$ and ${\mathrm{\partial }}_{j}z$ are continuous; Fubini [F6] makes $Z_j$ null there.

In the single reference box, write $q:=g-\mathcal{S}g$. Directly, $\mathrm{\parallel }q\mathrm{/}W{\mathrm{\parallel }}_{L^2}\le w_{-}^{-1}\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2}$ and $\mathrm{\parallel }\mathrm{\nabla }(q\mathrm{/}W){\mathrm{\parallel }}_{L^2}\le w_{-}^{-1}\mathrm{\parallel }\mathrm{\nabla }q{\mathrm{\parallel }}_{L^2}+G{w}_{-}^{-2}\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2}$ by the quotient rule and the stated lower and derivative bounds for $W$. Thus $\mathrm{\parallel }q\mathrm{/}W{\mathrm{\parallel }}_{H^1}\le (2\mathrm{/}{w}_{-}+G\mathrm{/}{w}_{-}^2)\mathrm{\parallel }q{\mathrm{\parallel }}_{H^1}\le {\widehat{\epsilon }}_h$ by [F3]. This is the direct reference-domain quotient bound.

Each partition face is contained in the image of an incident reference-cell face under its cell expression. The full cell derivative is nonsingular by admissibility, so its restriction to the $(d-1)$-dimensional face tangent space has rank $d-1$; since the rational cell expression extends $C^1$ across that closed face, the face is contained in the image of a $C^1$ map from an open subset of ${\mathbb{R}}^{d-1}$. By [F11] each such image, and hence each face, is Lebesgue null. Countably many boxes, finitely many cells, and these null faces show that $\mathrm{\nabla }z=0$ almost everywhere on $\{z=0\}$.

Choose a fixed $C^1$ function as follows: let $\chi (t)=0$ for $t\le 1$, $\chi (t)=3(t-1{)}^2-2(t-1{)}^3$ for $1<t<2$, and $\chi (t)=1$ for $t\ge 2$, and set $\theta (s)=s\chi (\mathrm{\mid }s\mathrm{\mid })$. Then $\theta =0$ on $[-1,1]$, $\theta (s)=s$ for $\mathrm{\mid }s\mathrm{\mid }\ge 2$, and $\mathrm{\parallel }{\theta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. Put $z_{ϵ}=ϵ\theta (z\mathrm{/}ϵ)$. Its cellwise chain rule and matching values give $z_{ϵ}\in H^1(D)$ by [F5], with gradient ${\theta }^{\mathrm{\prime }}(z\mathrm{/}ϵ)\mathrm{\nabla }z$. Its support lies in $K_{ϵ}:=\{x\in \bar{D}:\mathrm{\mid }z(x)\mathrm{\mid }\ge ϵ\}$, which is compact by [F9] and disjoint from $\mathrm{\partial }D$; hence $K_{ϵ}⋐D$ and [F5] gives $z_{ϵ}\in H_0^1(D)$. The function error is bounded by $C_{\theta }ϵ$ for $C_{\theta }:={\max }_{\mathrm{\mid }s\mathrm{\mid }\le 2}\mathrm{\mid }\theta (s)-s\mathrm{\mid }$, so its $L^2$ norm tends to zero because $\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$ by [F9]. The gradient error tends pointwise to zero off $\{z=0\}$, is zero almost everywhere on that set by step 2.1, and is bounded in squared magnitude by $(1+\mathrm{\parallel }{\theta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}{)}^2\mathrm{\mid }\mathrm{\nabla }z{\mathrm{\mid }}^2\in L^1(D)$. Dominated convergence [F6] proves $z_{ϵ}\to z$ in $H^1(D)$. Since $H_0^1(D)$ is closed by [F1], $z\in H_0^1(D)$. Applying this result to $v_h$ from step 1.2 gives $v_h\in V_h$.

Apply [F4] on each patch with numerator target $g_a$ and comparison $s_a$, whose required trace and cell regularity were checked in step 1.2. The denominator is continuous by the Statement; its positive cellwise polynomial expression and the closed-box C1 target $g_a$ make $g_a\mathrm{/}{W}_a$ C1 up to every cell boundary, and [A1] supplies Countable Choice. Then [F3] gives $\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_{H^1(D_a)}\le C_a{M}_a{B}_a$. The disjoint patch interiors cover $D$ up to their null faces; the global weak derivatives agree with the local derivatives by [F5]. Sum the squares of these H1 bounds to obtain $\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_{H^1(D)}\le {\epsilon }_h$. Step 1.1 then gives the projector error. Since $h_{a,r}\le 1$, the formula for $B_a$ is of order maximum span for fixed degrees; no neighboring-span ratio was used. Varying-degree convergence follows exactly when the displayed upper bound tends to zero.

Use $v_h\in V_h$ from step 3.1 in the best-gradient infimum of [F7] and bound its gradient error by its H1 error from step 4.1. This yields $(\beta \mathrm{/}\alpha ){\epsilon }_h$ for scalar elliptic data. Apply the same scalar construction to each displacement component; summing the component squared errors replaces $M_a^2$ by ${\sum }_q{M}_{a,q}^2$ and gives a conforming vector comparison. The elasticity branch of [F7] then yields the displayed finite-Lamé bound with its existing factor.

On the reference box let $w_h:=\mathcal{S}g\mathrm{/}W$. The same continuity, quotient and zero-boundary arguments of steps 1.2 and 3.1, with the identity as physical map, put $w_h$ in the reference rational space $V_h\subset H_0^1(\widehat{\mathrm{\Omega }})$. Step 1.4 proves $\mathrm{\parallel }u-w_h{\mathrm{\parallel }}_{H^1}\le {\widehat{\epsilon }}_h$. Apply step 1.1 to the reference-domain projector: its best-approximation property gives $\mathrm{\parallel }u-{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{H^1}\le \mathrm{\parallel }u-w_h{\mathrm{\parallel }}_{H^1}\le {\widehat{\epsilon }}_h$. In [F8] choose the admissible comparison ${\mathrm{\Pi }}_{h}u$ instead of taking the infimum; its gradient error is bounded by this H1 error. This proves the displayed Strang estimate without suppressing any defect. Finally [F8] bounds the geometry defect by the existing tensor constant times $\mathrm{\parallel }DF-DG{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{\mathrm{\nabla }}$, and step 1.1 gives $\mathrm{\parallel }{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{\mathrm{\nabla }}\le \mathrm{\parallel }{\mathrm{\Pi }}_{h}u{\mathrm{\parallel }}_{H^1}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}$. All original hypotheses of [F8] are retained.