Analytic data converge exponentially under fixed-mesh p refinement

On the reference cube define $I_p^{(j)}$ by applying [F2] in coordinate $j$, holding all other coordinates fixed, and put ${\mathcal{I}}_p=I_p^{(1)}\cdots I_p^{(d)}$. Each factor maps degree-at-most-$p$ polynomials in its coordinate to the same degree space, so ${\mathcal{I}}_p{\widehat{u}}_K\in {\mathbb{Q}}_p$. Fix a smooth one-dimensional section $f$ and write $g=f^{\mathrm{\prime }}$. By [F6], $f(x)-f(-1)$ is the Riemann integral of $g$ from $-1$ to $x$; [F7] identifies it with the Lebesgue integral used in [F2] and the coefficient formula [F2] (applied to the continuous products $g{P}_n$). Orthogonality [F5] makes the constant coefficient exact, so $I_{p}f$ preserves both endpoints. The bound $\mathrm{\mid }f(-1)\mathrm{\mid }\le 2^{-1\mathrm{/}2}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2}+\sqrt{2}\mathrm{\parallel }{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}$ follows by averaging $f(x)=f(-1)+{\int }_{-1}^x{f}^{\mathrm{\prime }}$ over $x$ and applying Cauchy–Schwarz [F4, F7]; together with $\mathrm{\parallel }{\mathrm{\Pi }}_{p-1}{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_2\le \mathrm{\parallel }{f}^{\mathrm{\prime }}{\mathrm{\parallel }}_2$ this gives $\mathrm{\parallel }{I}_{p}f{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }(I_{p}f{)}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}$ uniformly in $p$. Also, if an absolutely continuous $w$ vanishes at $-1$, then $w(x)={\int }_{-1}^x{w}^{\mathrm{\prime }}$ and Cauchy–Schwarz [F4] gives $\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2}\le 2\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}$.

Fix $j$ and a subset $S$ of the coordinate indices. If $j\in S$, then ${\mathrm{\partial }}_S(I_p^{(j)}-I){\widehat{u}}_K=({\mathrm{\Pi }}_{p-1}^{(j)}-I){\mathrm{\partial }}_j{\mathrm{\partial }}_{S\setminus \{j\}}{\widehat{u}}_K$. If $j\mathrm{\notin }S$, then ${\mathrm{\partial }}_S(I_p^{(j)}-I){\widehat{u}}_K=(I_p^{(j)}-I){\mathrm{\partial }}_S{\widehat{u}}_K$, whose $j$ derivative is $({\mathrm{\Pi }}_{p-1}^{(j)}-I){\mathrm{\partial }}_j{\mathrm{\partial }}_S{\widehat{u}}_K$ and whose value at $-1$ is zero. Thus in either case the error is controlled by the Legendre projection tail of $g={\mathrm{\partial }}_j{\mathrm{\partial }}_T{\widehat{u}}_K$, where $T=S\setminus \{j\}$ in the first case and $T=S$ in the second. Write $r=\mathrm{\mid }T\mathrm{\mid }\le d-1$ and let $T_{p-1}$ be its Taylor polynomial at zero through degree $p-1$. Taylor's Lagrange remainder [F3] bounds $\mathrm{\mid }g-T_{p-1}\mathrm{\mid }$ on $[-1,1]$ by $M(p+r+1)!R^{-p-r-1}\mathrm{/}p!$. Orthogonality of ${\mathrm{\Pi }}_{p-1}^{(j)}$ gives $\mathrm{\parallel }g-T_{p-1}{\mathrm{\parallel }}_2^2=\mathrm{\parallel }g-{\mathrm{\Pi }}_{p-1}^{(j)}g{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{\mathrm{\Pi }}_{p-1}^{(j)}g-T_{p-1}{\mathrm{\parallel }}_2^2$, so the projection error is at most $\sqrt{2}M(p+r+1)!R^{-p-r-1}\mathrm{/}p!$. For fixed $d,R,q$ this is at most $C_{d,R,q}M{q}^p$: the factorial ratio is bounded by a fixed-degree polynomial in $p$, while $(Rq{)}^{-p}$ decays geometrically. Anchoring at $-1$ and the one-dimensional Poincare estimate from step 1.1 use [F6, F7] for the continuous error derivative and give the same bound for each mixed derivative of $(I_p^{(j)}-I){\widehat{u}}_K$.

The estimate in step 1.1 is stable in every mixed derivative: if a mixed derivative includes coordinate $j$, differentiating $I_p^{(j)}$ there gives the contractive projection ${\mathrm{\Pi }}_{p-1}^{(j)}$; if it omits $j$, the one-dimensional $H^1$ bound in step 1.1 controls the application of $I_p^{(j)}$ using that derivative and its additional $j$ derivative. Summing over the finitely many subsets $S$ gives $\mathrm{\parallel }{I}_p^{(j)}f{\mathrm{\parallel }}_{\mathrm{m}\mathrm{i}\mathrm{x}}\le C_d\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ uniformly in $p$. The telescoping identity ${\mathcal{I}}_p-I={\sum }_{j=1}^d{I}_p^{(1)}\cdots I_p^{(j-1)}(I_p^{(j)}-I)$ and step 2.1 now imply $\mathrm{\parallel }{\widehat{u}}_K-{\mathcal{I}}_p{\widehat{u}}_K{\mathrm{\parallel }}_{H^1([-1,1{]}^d)}\le C_{d,R,q}M{q}^p$.

The one-dimensional projector preserves both endpoints. It commutes with reflection: both the reflected polynomial and the projection of the reflected function have the same derivative, since reflection preserves the polynomial subspace and the $L^2$ inner product, and they have the same value at the left endpoint by preservation of the right endpoint. Coordinate projectors commute under coordinate permutations. On a shared box face the relative affine parameter map is an affine automorphism of the reference cube face: it permutes the supporting facet hyperplanes and preserves their vertices, so its linear part is a signed coordinate permutation. Thus the restrictions of the transported tensor projectors are the same tangential projector applied to the common trace of $u$, and the resulting cell traces match. The conformity result [F8] gives $u_p\in H^1(\mathrm{\Omega })$; since $u\in H^1(\mathrm{\Omega })$, the local estimates sum to the stated global $H^1$ estimate. Scaling from each reference cell to its physical affine box multiplies the local estimate by a fixed cell-dependent factor; the mesh has finitely many cells, so taking the largest factor gives the global constant $C_{\mathcal{T},d,R,q}$.