%\path[draw, arr] (X3) -- (X3h) node [midway, anchor=west, yshift=0.8mm] {$\Pibr^3$};
\end{tikzpicture}
%%\includegraphics{img/de_rham/de_rham_2D_grad_rot_cont}\fsp,
\end{split}$$ Both the continuous and the discrete sequences in  and  are exact for simply connected domains [@buffa:2011; @buffa:2013]. Moreover we assume that the following estimates hold: $$\label{eq:app_est_2D_cart}
\!\!\!\begin{alignedat}{3}
& \norm{u - \Pi^0 u}_{\Hsp^1(\srfdomflat)}  &&\leq C
h^{s}\norm{u}_{\Hsp^{s+1}(\srfdomflat)}~, &&\quad u \in
\Hsp^{s+1}(\srfdomflat)~,\\ 
& \norm{\uv - \Pi^1 \uv}_{\Hsp(\curltsc;\,\srfdomflat)}&&\leq C
h^{s}\norm{\uv}_{\Hsp^{s}(\curltsc;\,\srfdomflat)}~, &&\quad \uv \in
\Hsp^{s}(\curltsc;\,\srfdomflat)~,\\ 
&\norm{\uv - \Pi^{1*} \uv}_{\Hsp(\divergt;\,\srfdomflat)} &&\leq C
h^{s}\norm{\uv}_{\Hsp^{s}(\divergt;\,\srfdomflat)}~, &&\quad \uv \in
\Hsp^{s}(\divergt;\,\srfdomflat)~,\\ 
&\norm{u - \Pi^2 u}_{\Lsp^2(\srfdomflat)} &&\leq C
h^{s}\norm{u}_{\Hsp^{s}(\srfdomflat)}~, &&\quad u \in
\Hsp^{s}(\srfdomflat).
\end{alignedat}\!$$

In [@oh:2015], the discrete Cartesian spaces were chosen as piecewise linear for $\CaSpB_h^0$, lowest order Nedelec for $\CaSpB_h^1$, lowest order Raviart-Thomas for $\CaSpB_h^{1*}$ and piecewise constant for $\CaSpB_h^2$. In this work, we will use spaces that will be introduced in the next section.

The strategy to define the discrete counterparts of  is based on the use of operators that link the spaces in cylindrical coordinates to the standard spaces in Cartesian coordinates. The definition of the discrete spaces and of the projectors, together with error estimates, are then deduced from those defined for the spaces in Cartesian coordinates. The diagram $$\label{eq:gen_strategy}
\begin{split}
\tikzsetnextfilename{de_rham/de_rham_3d_cyl_gen_strat}
\begin{tikzpicture}[commutative diagrams/every diagram,
arr/.style={commutative diagrams/.cd,
    every arrow,
    every label}]
\def\minW{25mm}
\def\minH{18mm}
