approximation properties of the smoothed projection operators of the finite element spaces [@AFW].

\[SmoothPr\][ *There exist linear projection operators $$\begin{aligned}
&\widetilde\Pi_h^{\mathbb S}: {\cal L}^1\rightarrow {\mathbb S}_h^r,
\qquad\quad
\widetilde\Pi_h^{\mathbb V}: L^1\rightarrow {\mathbb V}_h^{k+1},
\qquad\quad
\widetilde\Pi_h^{\mathbb N}: {\bf L}^1\rightarrow {\mathbb N}_h^k, \end{aligned}$$ which satisfy $$\begin{aligned}
&\nabla (\widetilde\Pi_h^{\mathbb V}\chi)
=\widetilde\Pi_h^{\mathbb N} \nabla \chi,
&&\forall\,\chi\in W^{1,1}  ,\\
&\|\varphi-\widetilde\Pi_h^{\mathbb S}\varphi\|_{{\cal L}^p}
\leq Ch^{s+3/p-3/q}\|\varphi\|_{{\cal W}^{s,q}}, &&
\forall\,\varphi\in {\cal W}^{s,q},\,\,\,\,\,\,
0\leq s\leq r+1, \\
&\|\chi-\widetilde\Pi_h^{\mathbb V}\chi\|_{L^p}
\leq Ch^{s+3/p-3/q}\|\chi\|_{W^{s,q}}, &&
\forall\,\chi\in W^{s,q},\,\,\,\,\,\,
0\leq s\leq k+2, \\
&\|{\bf a}-\widetilde\Pi_h^{\mathbb N}{\bf a}\|_{{\bf L}^p}
\leq Ch^{s+3/p-3/q}\|{\bf a}\|_{{\bf W}^{s,q}}, 
&&\forall\,{\bf a}\in {\bf W}^{s,q},\,\,\,\,\,\,
0\leq s\leq k+1, \end{aligned}$$ for any $$\left\{\begin{array}{ll}
1\leq q\leq p\leq 3/(3/q-s) &\mbox{if}\,\,\,\,\, 0\leq s< 3/q,\\
1\leq q\leq p<\infty &\mbox{if}\,\,\,\,\, s\geq 3/q .
\end{array}
\right.$$* ]{}

The authors of [@AFW] (page 66–70) only proved the $L^2$ boundedness of the smoothed projection operators. But their method can also be used to prove the $L^p$ boundedness without essential change. Then Lemma \[SmoothPr\] is obtained by using the Sobolev embedding $W^{s,q}\hookrightarrow W^{s+3/p-3/q,p}$. Although the analysis of [@AFW] (page 66–70) only considered polyhedra, the extension to curved polyhedra is straightforward (as there are no boundary conditions imposed on these finite element spaces).

It is well known that the solution of the heat equation $$\begin{aligned}
