Local interpolation estimates on shape-regular affine meshes

On the reference cell, write $\widehat{I}w$ as the finite sum of its degree-of-freedom values times the dual shape functions. The assumed boundedness of each functional and [F2] bound $\mathrm{\mid }\widehat{I}w{\mathrm{\mid }}_{W^{m,p}}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{W^{\mathrm{\ell },p}}$. For any reproduced $q$, linearity gives $\widehat{v}-\widehat{I}\widehat{v}=(\widehat{v}-q)-\widehat{I}(\widehat{v}-q),$ so $\mathrm{\mid }\widehat{v}-\widehat{I}\widehat{v}{\mathrm{\mid }}_{W^{m,p}}\le \mathrm{\mid }\widehat{v}-q{\mathrm{\mid }}_{W^{m,p}}+C\mathrm{\parallel }\widehat{v}-q{\mathrm{\parallel }}_{W^{\mathrm{\ell },p}}\mathrm{.}$ For the automatic Lagrange application, under the additional Axiom of Choice, $d\ge 2$, and $\mathrm{\ell }>d\mathrm{/}p$, [F6] verifies nodal-functional boundedness. For the generic estimate, boundedness is the explicit hypothesis, so this additional choice assumption is not used.

If $\mathrm{\ell }\le r\le k+1$ and $\widehat{v}$ is real-valued, choose the Bramble–Hilbert polynomial $q\in {\mathbb{P}}_{r-1}\subseteq {\mathbb{P}}_k$ from [F3]. If $\widehat{v}$ is complex-valued, apply the first clause of [F3] to $\mathrm{Re}\widehat{v}$ and $\mathrm{Im}\widehat{v}$ separately, obtaining $q_R,q_I\in {\mathbb{P}}_{r-1}$, and set $q=q_R+i{q}_I$. In either case $q$ is reproduced by $\widehat{I}$. Since $m\le r$ and $\mathrm{\ell }\le r$, [F3] bounds both terms in step 1.1 for the real case; for the complex case, [F8] combines the two real estimates and changes only the constant. Thus $\mathrm{\mid }\widehat{v}-\widehat{I}\widehat{v}{\mathrm{\mid }}_{W^{m,p}}\le C\mathrm{\mid }\widehat{v}{\mathrm{\mid }}_{W^{r,p}}$.

If $\mathrm{\ell }>k+1$ and $\widehat{v}$ is real-valued, choose the Bramble–Hilbert polynomial $q\in {\mathbb{P}}_k$ from [F3]. If $\widehat{v}$ is complex-valued, apply the second clause of [F3] with $r=k+1$, $s=\mathrm{\ell }$ to its real and imaginary parts, obtaining $q_R,q_I\in {\mathbb{P}}_k$, and set $q=q_R+i{q}_I$. In either case $q$ is reproduced by $\widehat{I}$. Then [F3] bounds the $W^{\mathrm{\ell },p}$ norm of $\widehat{v}-q$ by $C{\sum }_{n=k+1}^{\mathrm{\ell }}\mathrm{\mid }\widehat{v}{\mathrm{\mid }}_{W^{n,p}}$; because $m\le k+1<\mathrm{\ell }$, the same controls its $W^{m,p}$ norm. For complex $\widehat{v}$, [F8] combines the component estimates with at most a fixed change in $C$. Step 1.1 gives $\mathrm{\mid }\widehat{v}-\widehat{I}\widehat{v}{\mathrm{\mid }}_{W^{m,p}}\le C{\sum }_{n=k+1}^{\mathrm{\ell }}\mathrm{\mid }\widehat{v}{\mathrm{\mid }}_{W^{n,p}}\mathrm{.}$

For $\widehat{v}=v\circ F_K$, use [F4] to transform the physical error to the reference cell and the reference derivatives of $v$ back to the physical cell. The product of the two determinant factors cancels; the remaining matrix factors are bounded by $C\mathrm{\parallel }{A}_K^{-1}{\mathrm{\parallel }}^m\mathrm{\parallel }{A}_K{\mathrm{\parallel }}^r\le C{h}_K^{r-m}$ by [F5]. This gives the first physical estimate from step 2.1.

In the second case, apply the same transformation separately to each order $n$ in step 2.2. Its matrix factor is bounded by $C\mathrm{\parallel }{A}_K^{-1}{\mathrm{\parallel }}^m\mathrm{\parallel }{A}_K{\mathrm{\parallel }}^n\le C{h}_K^{n-m}$ by [F5]; summing over $k+1\le n\le \mathrm{\ell }$ gives the second physical estimate. The constants in [F2]–[F7] are uniform over the fixed reference element and the shape-regular family, proving uniformity.