Constructive varying-degree triangle interpolation in unweighted H1

Justify the Sobolev edge assertion in [F4] from the actual $H^1$ trace theorem, rather than assuming a stronger trace statement. On the reference triangle choose smooth $w_j\to w$ in $H^2$ by its smooth-density supplier. The $H^1$ trace theorem applied to $w_j$ and separately to each ${\mathrm{\partial }}_i{w}_j\in H^1$ gives ${\gamma }_E{w}_j\to {\gamma }_{E}w$ and ${\gamma }_E\mathrm{\nabla }{w}_j\to {\gamma }_E\mathrm{\nabla }w$ in $L^2(E)$. For every $\zeta \in C_c^{\mathrm{\infty }}(E)$, ordinary one-dimensional integration by parts gives ${\int }_E{\gamma }_E{w}_j{\mathrm{\partial }}_{\tau }\zeta =-{\int }_E({\gamma }_E\mathrm{\nabla }{w}_j\cdot \tau )\zeta$. Passing to the $L^2$ limits proves ${\mathrm{\partial }}_{\tau }{\gamma }_{E}w={\gamma }_E\mathrm{\nabla }w\cdot \tau$ in distributions and hence ${\gamma }_{E}w\in H^1(E)$, with norm at most $C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^2}$. Its continuous edge representative agrees with the continuous representative of $w$: smooth approximants converge uniformly on the edge by the one-dimensional $H^1$ bound [F6], while the point-evaluation supplier in [F4] gives their convergence to $w$ at each fixed edge point, including the vertices. Transporting the value trace and the gradient trace separately to $K$ yields $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(E)}\le C_{\gamma }(h_K^{-1\mathrm{/}2}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(K)}+h_K^{1\mathrm{/}2}\mathrm{\mid }w{\mathrm{\mid }}_{H^1(K)})$ and $\mathrm{\parallel }{\mathrm{\partial }}_{\tau }w{\mathrm{\parallel }}_{L^2(E)}\le C_{\gamma }(h_K^{-1\mathrm{/}2}\mathrm{\mid }w{\mathrm{\mid }}_{H^1(K)}+h_K^{1\mathrm{/}2}\mathrm{\mid }w{\mathrm{\mid }}_{H^2(K)})$. Thus the scaled edge expression in the Statement is bounded by $C_{\gamma }(h_K^{-1}\mathrm{\parallel }w{\mathrm{\parallel }}_2+\mathrm{\mid }w{\mathrm{\mid }}_{H^1}+h_K\mathrm{\mid }w{\mathrm{\mid }}_{H^2})$, precisely the scaled $H^2$ quantity used in the final rate.

Fix a single orientation of every geometric edge and affine-transport the operator in [F1] to define $g_E=I_{p_E}(u{\mathrm{\mid }}_E)$. For any $s\in {\mathbb{P}}_{p_E}(E)$, its derivative lies in ${\mathbb{P}}_{p_E-1}(E)$, so the projection is exact and $I_{p_E}s=s$. For a general absolutely continuous $f$ on $E$, the residual $f^{\mathrm{\prime }}-{\mathrm{\Pi }}_{p_E-1}{f}^{\mathrm{\prime }}$ is orthogonal to constants, so its integral over $E$ is zero; the anchor formula and [F6] therefore show that $I_{p_E}f$ has both endpoint values of $f$. In particular $g_E$ takes the endpoint values of $u$. For each cell $K$, let $a_K$ be the affine polynomial with the three vertex values of $u$. Each residual $d_E=g_E-a_K{\mathrm{\mid }}_E$ has zero endpoints and degree at most $p_E\le p_K$. Sum the three lifts from [F2] and put $G_K=a_K+{\sum }_{E\subset \mathrm{\partial }K}{L}_E{d}_E$; it has degree at most $p_K$ and trace $g_E$ on each edge. Let $Z_K=\{z\in {\mathbb{P}}_{p_K}(K):z{\mathrm{\mid }}_{\mathrm{\partial }K}=0\}$. In the finite-dimensional affine space $G_K+Z_K$, choose the unique $Iu{\mathrm{\mid }}_K$ satisfying ${\int }_K\mathrm{\nabla }(Iu-u)\cdot \mathrm{\nabla }z=0$ for every $z\in Z_K$. Existence and uniqueness follow from the positive-definite Gram matrix of the gradients on $Z_K$: if ${\int }_K\mathrm{\mid }\mathrm{\nabla }z{\mathrm{\mid }}^2=0$, the polynomial $z$ is constant, and its zero boundary trace forces $z=0$. If $Z_K=\{0\}$, the system is empty and $Iu{\mathrm{\mid }}_K=G_K$. Thus the construction is linear and uses only uniquely determined finite systems.

Both cells incident to an edge use the same oriented edge polynomial $g_E$, so their traces agree and have degree at most $p_E$. By [F5], the assembled interpolant is in the trace-matching $H^1$ space. If $E$ is a boundary edge on which $u=0$, then $g_E=I_{p_E}0=0$, so the interpolant also vanishes there.

Fix $q\in {\mathbb{P}}_{m_K}(K)$ and put $w=u-q$. Since $m_K\le p_E$, [F1] reproduces $q{\mathrm{\mid }}_E$, so $g_E-q{\mathrm{\mid }}_E=I_{p_E}(w{\mathrm{\mid }}_E)$. Its endpoint values are those of $w$. The one-dimensional endpoint estimate from [F4] bounds each vertex value $\mathrm{\mid }w(v)\mathrm{\mid }$ by $C({\mathrm{\ell }}_E^{-1\mathrm{/}2}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(E)}+{\mathrm{\ell }}_E^{1\mathrm{/}2}\mathrm{\parallel }{\mathrm{\partial }}_{{\tau }_E}w{\mathrm{\parallel }}_{L^2(E)})$ on any incident edge. Consequently the affine interpolant $a_K^w$ of the vertex values has $\mathrm{\mid }{a}_K^w{\mathrm{\mid }}_{H^1(K)}+h_K^{-1}\mathrm{\parallel }{a}_K^w{\mathrm{\parallel }}_{L^2(K)}$ bounded by the sum of these three edge terms. On each edge, $I_{p_E}(w{\mathrm{\mid }}_E)-a_K^w{\mathrm{\mid }}_E$ vanishes at its endpoints. The derivative contraction in [F1] and the preceding endpoint estimate bound its strong edge norm by the same sum. Applying [F2] to the three endpoint-zero residuals and adding the lifts gives $c_K\in {\mathbb{P}}_{p_K}(K)$ with trace $g_E-q{\mathrm{\mid }}_E$ and $\mathrm{\mid }{c}_K{\mathrm{\mid }}_{H^1(K)}+h_K^{-1}\mathrm{\parallel }{c}_K{\mathrm{\parallel }}_{L^2(K)}\le C_{\gamma }{\sum }_{E\subset \mathrm{\partial }K}({\mathrm{\ell }}_E^{-1\mathrm{/}2}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(E)}+{\mathrm{\ell }}_E^{1\mathrm{/}2}\mathrm{\parallel }{\mathrm{\partial }}_{{\tau }_E}w{\mathrm{\parallel }}_{L^2(E)})\mathrm{.}$

The polynomial $q+c_K$ has the same boundary trace as $Iu{\mathrm{\mid }}_K$, so $z:=q+c_K-Iu\in Z_K$. The energy orthogonality in step 1.2 makes $Iu$ the best $H^1$-seminorm approximation to $u$ in the affine space $G_K+Z_K$; hence $\mathrm{\mid }u-Iu{\mathrm{\mid }}_{H^1(K)}\le \mathrm{\mid }u-q-c_K{\mathrm{\mid }}_{H^1(K)}\le \mathrm{\mid }u-q{\mathrm{\mid }}_{H^1(K)}+\mathrm{\mid }{c}_K{\mathrm{\mid }}_{H^1(K)}$. This and step 2.2 prove the first estimate. For the full norm, the reference-triangle identity $z(x,y)={\int }_0^x{\mathrm{\partial }}_{x}z(t,y)dt$ holds because $z=0$ on $x=0$; Fubini and Cauchy–Schwarz [F6] give $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^2(K)}\le C_{\gamma }{h}_K\mathrm{\mid }z{\mathrm{\mid }}_{H^1(K)}$. Also $\mathrm{\mid }z{\mathrm{\mid }}_{H^1(K)}\le \mathrm{\mid }q+c_K-u{\mathrm{\mid }}_{H^1(K)}+\mathrm{\mid }u-Iu{\mathrm{\mid }}_{H^1(K)}\le 2(\mathrm{\mid }u-q{\mathrm{\mid }}_{H^1(K)}+\mathrm{\mid }{c}_K{\mathrm{\mid }}_{H^1(K)})$. Decompose $u-Iu=(u-q)-c_K+z$, use the triangle inequality and the $L^2$ lift bound from step 2.2, and obtain the displayed full-norm estimate. If $u{\mathrm{\mid }}_K\in H^r(K)$, choose $q=A_{m_K,r}u$ from [F3]. The proved edge trace bounds in step 1.1 control every edge term by $C_{r,\gamma }{h}_K^{r-1}{\eta }_{n_K,r}\mathrm{\mid }u{\mathrm{\mid }}_{H^r(K)}$, proving the stated rate. Finally ${\eta }_{n_K,r}^{-2}$ is the product of $2(r-4)$ consecutive integers beginning at $n_K-r+4\ge m_K\mathrm{/}4$; taking square roots proves the simplified power bound.