Appendix
1.1 The Proof of Theorem 3.5
First, we concentrate on the case \(k=1\). From Lemma 3.3, \(\Pi _{K_i}\varvec{E}\,(i=1,\ldots ,4)\) can be obtained. Now let us use the the recovery method (3.1) for \(\Pi _{K_i}\varvec{E}\). We assume that \(\varvec{R}[\Pi _{K_i}\varvec{E}] \in Q_{0,1}(K_{s})\times Q_{1,0}(K_{s})\) has the following form:
$$\begin{aligned} \varvec{R}[\Pi _{K_i}\varvec{E}]=a\varvec{\phi }_{u,K_{s}}^{0,0} +b\varvec{\phi }_{u,K_{s}}^{0,1} +c\varvec{\phi }_{v,K_{s}}^{0,0} +d\varvec{\phi }_{u,K_{s}}^{1,0}. \end{aligned}$$
Equivalently, it forms the following linear equations:
$$\begin{aligned} \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} \frac{16}{3} &{} \frac{8}{3} &{} 0 &{} 0\\ \frac{8}{3} &{} \frac{16}{3} &{} 0 &{} 0\\ 0 &{} 0 &{} \frac{16}{3} &{} \frac{8}{3} \\ 0 &{} 0 &{} \frac{8}{3} &{} \frac{16}{3} \end{array} \right] \left[ \begin{array}{lll} a \\ b\\ c\\ d \end{array} \right] = \left[ \begin{array}{ccc} \sum \limits _{i=1}^{4}\int _{K_{s}^{i}} \Pi _{K_i}\varvec{E}\cdot \varvec{\phi }_{u,K_s}^{0,0}d K \\ \sum \limits _{i=1}^{4}\int _{K_{s}^{i}} \Pi _{K_i}\varvec{E}\cdot \varvec{\phi }_{u,K_s}^{0,1}d K \\ \sum \limits _{i=1}^{4}\int _{K_{s}^{i}} \Pi _{K_i}\varvec{E}\cdot \varvec{\phi }_{v,K_s}^{0,0}d K \\ \sum \limits _{i=1}^{4}\int _{K_{s}^{i}} \Pi _{K_i}\varvec{E}\cdot \varvec{\phi }_{v,K_s}^{1,0}d K \end{array} \right] =: \left[ \begin{array}{lll} r_{1} \\ r_{2} \\ r_{3} \\ r_{4} \end{array} \right] , \end{aligned}$$
(5.1)
where
$$\begin{aligned} r_{1}= & {} \frac{10}{3}s^{2}\left( c_{u,K_{1}}^{0,0}+c_{u,K_{2}}^{0,0}\right) + \left( 6s-\frac{10}{3}s^2\right) \left( c_{u,K_{1}}^{0,1}+c_{u,K_{2}}^{0,1}\right) \\&+ \left( 2s-\frac{2}{3}s^2\right) \left( c_{u,K_3}^{0,0}+c_{u,K_4}^{0,0}\right) + \frac{2}{3}s^{2}\left( c_{u,K_3}^{0,1}+c_{u,K_4}^{0,1}\right) , \\ r_{2}= & {} \frac{2}{3}s^{2}\left( c_{u,K_{1}}^{0,0}+c_{u,K_{2}}^{0,0}\right) +\left( 2s-\frac{2}{3}s^2\right) \left( c_{u,K_{1}}^{0,1}+c_{u,K_{2}}^{0,1}\right) \\&+ \left( 6s-\frac{10}{3}s^2\right) \left( c_{u,K_3}^{0,0}+c_{u,K_4}^{0,0}\right) + \frac{10}{3}s^{2}\left( c_{u,K_{3}}^{0,1}+c_{u,K_{4}}^{0,1}\right) , \\ r_{3}= & {} \frac{10}{3}s^{2}\left( c_{v,K_{1}}^{0,0}+c_{v,K_{3}}^{0,0}\right) +\left( 6s-\frac{10}{3}s^2\right) \left( c_{v,K_{1}}^{1,0}+c_{v,K_{3}}^{1,0}\right) \\&+ \left( 2s-\frac{2}{3}s^2\right) \left( c_{v,K_{2}}^{0,0}+c_{v,K_{4}}^{0,0}\right) +\frac{2}{3}s^{2}\left( c_{v,K_{2}}^{1,0}+c_{v,K_{4}}^{1,0}\right) , \\ r_{4}= & {} \frac{2}{3}s^{2}\left( c_{v,K_{1}}^{0,0}+c_{v,K_{3}}^{0,0}\right) +\left( 2s-\frac{2}{3}s^2\right) \left( c_{v,K_{1}}^{1,0}+c_{v,K_{3}}^{1,0}\right) \\&+ \left( 6-s\frac{10}{3}s^2\right) \left( c_{v,K_{2}}^{0,0}+c_{v,K_{4}}^{0,0}\right) +\frac{10}{3}s^{2}\left( c_{v,K_{2}}^{1,0}+c_{v,K_{4}}^{1,0}\right) . \end{aligned}$$
By solving (5.1), it is easy to obtain:
$$\begin{aligned}&a=\frac{1}{8}(2r_{1}-r_{2}), \,\quad b=\frac{1}{8}(-r_{1}+2r_{2}), \end{aligned}$$
(5.2)
$$\begin{aligned}&c=\frac{1}{8}(2r_{3}-r_{4}),\quad \,d=\frac{1}{8}(-r_{3}+2r_{4}). \end{aligned}$$
(5.3)
Then, we have:
$$\begin{aligned}&\varvec{R}[\Pi _{K_i}\varvec{E}](O) = \left[ \begin{array}{ccc} \frac{1}{sh}(a+b) \\ \frac{1}{sh}(c+d) \end{array} \right] =: \left[ \begin{array}{ccc} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_1 \\ (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_2 \end{array}\right] . \end{aligned}$$
(5.4)
From (5.2), the first component is as follows:
$$\begin{aligned} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_1= & {} \frac{1}{8sh}(r_{1}+r_{2}) \nonumber \\= & {} \frac{1}{8sh}\left[ 4s^{2}\left( c_{u,K_{1}}^{0,0}+c_{u,K_{2}}^{0,0}\right) + \left( 8-4s\right) s\left( c_{u,K_{1}}^{0,1}+c_{u,K_{2}}^{0,1}\right) \nonumber \right. \\&\left. + \left( 8-4s\right) s\left( c_{u,K_3}^{0,0}+c_{u,K_4}^{0,0}\right) + 4s^{2}\left( c_{u,K_3}^{0,1}+c_{u,K_4}^{0,1}\right) \right] . \end{aligned}$$
(5.5)
From Lemma 3.3 and the second order Taylor expansion of \(E_{1}\) at O:
$$\begin{aligned} E_{1}=E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +y\frac{\partial E_{1}}{\partial y}(O)+O(h^{2})\Vert \partial ^{2}E_{1}\Vert _{\infty }, \end{aligned}$$
we obtain
$$\begin{aligned} c_{u,K_{1}}^{0,0}= & {} \frac{1}{4}\int _{-h}^{0}\left[ E_{1}(O) +x\frac{\partial E_{1}}{\partial x}(O) +(-h)\frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.6)
$$\begin{aligned} c_{u,K_{1}}^{0,1}= & {} \frac{1}{4}\int _{-h}^{0}\left[ E_{1}(O) +x\frac{\partial E_{1}}{\partial x}(O) +0*\frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.7)
$$\begin{aligned} c_{u,K_{2}}^{0,0}= & {} \frac{1}{4}\int _{0}^{h} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +(-h)\frac{\partial u_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.8)
$$\begin{aligned} c_{u,K_{2}}^{0,1}= & {} \frac{1}{4}\int _{0}^{h} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +0\times \frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.9)
$$\begin{aligned} c_{u,K_{3}}^{0,0}= & {} \frac{1}{4}\int _{-h}^{0} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +0\times \frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.10)
$$\begin{aligned} c_{u,K_{3}}^{0,1}= & {} \frac{1}{4}\int _{-h}^{0} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +h\times \frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.11)
$$\begin{aligned} c_{u,K_{4}}^{0,0}= & {} \frac{1}{4}\int _{0}^{h} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +0\times \frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty },\qquad \end{aligned}$$
(5.12)
$$\begin{aligned} c_{u,K_{4}}^{0,1}= & {} \frac{1}{4}\int _{0}^{h} \left[ E_{1}(O)+x\frac{\partial E_{1}}{\partial x}(O) +h\times \frac{\partial E_{1}}{\partial y}(O)\right] dx+O(h^{3})\Vert \partial ^{2}E_{1}\Vert _{\infty }.\qquad \end{aligned}$$
(5.13)
Substituting (5.6)–(5.13) into (5.5), we have
$$\begin{aligned} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_1 =E_{1}(O)+O(h^{2})\Vert \partial ^{2}E_{1}\Vert _{\infty }. \end{aligned}$$
(5.14)
Similarly, for the second component, we have
$$\begin{aligned} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_2 =E_{2}(O)+O(h^{2})\Vert \partial ^{2}E_{2}\Vert _{\infty }. \end{aligned}$$
(5.15)
From (5.14) and (5.15), we find that the value of \(\varvec{R}[\Pi _{K_i}\varvec{E}]\) at O is the second order accuracy for every \(s\in (0,1]\).
Next, we deal with the case \(k=2\). From Lemma 3.4, we already know that the Nédélec interpolation \(\Pi _{K_i}\varvec{E}\). Now let us use the the recovery method (3.1) for \(\Pi _{K_i}\varvec{E}\). Let us assume \(\varvec{R}[\Pi _{K_i}\varvec{E}]\in Q_{1,2}(K_s)\times Q_{2,1}(K_s)\) as follows:
$$\begin{aligned} \varvec{R}[\Pi _{K_i}\varvec{E}]=\sum _{j=0}^{1}\sum _{k=0}^{2} c_{u,K_s}^{j,k}\varvec{\phi }_{u,K_s}^{j,k} +\sum _{j=0}^{1}\sum _{k=0}^{2} c_{v,K_s}^{k,j} \varvec{\phi }_{v,K_s}^{k,j}. \end{aligned}$$
Equivalently, it follows that
$$\begin{aligned} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \varvec{A}_{\varvec{u}} &{} O \\ O &{} \varvec{A}_{\varvec{v}} \end{array} \right] \left[ \begin{array}{ccc} \varvec{c}_{\varvec{u}} \\ \varvec{c}_{\varvec{v}} \end{array} \right] =\left[ \begin{array}{ccc} \varvec{r}_{\varvec{u}} \\ \varvec{r}_{\varvec{v}} \end{array} \right] , \end{aligned}$$
(5.16)
where
$$\begin{aligned} A_{\varvec{u}}=A_{\varvec{v}}= \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \frac{16}{3} &{} 0 &{} \frac{8}{3} &{} 0 &{} \frac{-8}{3} &{} 0 \\ 0 &{} \frac{16}{9} &{} 0 &{} \frac{8}{9} &{} 0 &{} \frac{-8}{9} \\ \frac{8}{3} &{} 0 &{} \frac{16}{3} &{} 0 &{} \frac{-8}{3} &{} 0 \\ 0 &{} \frac{8}{9} &{} 0 &{} \frac{16}{9} &{} 0 &{} \frac{-8}{9} \\ \frac{-8}{3} &{} 0 &{} \frac{-8}{3} &{} 0 &{} \frac{32}{15}&{} 0 \\ 0 &{} \frac{-8}{9} &{} 0 &{} \frac{-8}{9} &{} 0 &{} \frac{32}{45} \end{array} \right] , \end{aligned}$$
(5.17)
$$\begin{aligned}&\varvec{c}_{\varvec{u}}^{T}=\left[ \begin{array}{ccc} c_{u,K_s}^{0,0},\quad c_{u,K_s}^{1,0},\quad c_{u,K_s}^{0,1},\quad c_{u,K_s}^{1,1},\quad c_{u,K_s}^{0,2},\quad c_{u,K_s}^{1,2} \end{array} \right] , \\&\varvec{c}_{\varvec{v}}^{T}=\left[ \begin{array}{lll} c_{v,K_s}^{0,0},\quad c_{v,K_s}^{0,1},\quad c_{v,K_s}^{1,0},\quad c_{v,K_s}^{1,1},\quad C_{v,K_s}^{2,0},\quad a_{v,K_s}^{2,1} \end{array} \right] ,\quad \\&\varvec{r}_{\varvec{u}}^{T}=\left[ \begin{array}{ccc} r_{u,K_s}^{0,0},\quad r_{u,K_s}^{1,0},\quad r_{u,K_s}^{0,1},\quad r_{u,K_s}^{1,1},\quad r_{u,K_s}^{0,2},\quad r_{u,K_s}^{1,2} \end{array} \right] , \\&\varvec{r}_{\varvec{v}}^{T}=\left[ \begin{array}{lll} r_{v,K_s}^{0,0},\quad r_{v,K_s}^{0,1},\quad r_{v,K_s}^{1,0},\quad r_{v,K_s}^{1,1},\quad r_{v,K_s}^{2,0},\quad r_{v,K_s}^{2,1} \end{array} \right] , \end{aligned}$$
with
$$\begin{aligned} r_{u,K_s}^{j,k}= \sum _{i=1}^{4}\int _{K_s^i}\Pi _{K_i}\varvec{E}\cdot \varvec{\phi }_{u,K_s}^{j,k}dK,\quad r_{v,K_s}^{k,j}= \sum _{i=1}^{4}\int _{K_s^i}\Pi _{K_i}\varvec{E} \cdot \varvec{\phi }_{v,K_s}^{k,j}dK. \end{aligned}$$
(5.18)
By solving the linear system (5.16), we obtain
$$\begin{aligned} c_{u,K_s}^{0,0}= & {} \frac{3}{16}\left( 3r_{u,K_s}^{0,0}+r_{u,K_s}^{0,1}+5r_{u,K_s}^{0,2}\right) , \end{aligned}$$
(5.19)
$$\begin{aligned} c_{u,K_s}^{0,1}= & {} \frac{3}{16}\left( r_{u,K_s}^{0,0}+3r_{u,K_s}^{0,1}+5r_{u,K_s}^{0,2}\right) , \end{aligned}$$
(5.20)
$$\begin{aligned} c_{u,K_s}^{0,2}= & {} \frac{15}{16}\left( r_{u,K_s}^{0,0}+r_{u,K_s}^{0,1} +3r_{u,K_s}^{0,2}\right) ,\, \end{aligned}$$
(5.21)
$$\begin{aligned} c_{u,K_s}^{1,0}= & {} \frac{9}{16}\left( 3r_{u,K_s}^{1,0}+r_{u,K_s}^{1,1} +5r_{u,K_s}^{1,2}\right) ,\, \end{aligned}$$
(5.22)
$$\begin{aligned} c_{u,K_s}^{1,1}= & {} \frac{9}{16}\left( r_{u,K_s}^{1,0} +3r_{u,K_s}^{1,1}+5r_{u,K_s}^{1,2}\right) ,\, \end{aligned}$$
(5.23)
$$\begin{aligned} c_{u,K_s}^{1,2}= & {} \frac{45}{16}\left( r_{u,K_s}^{1,0} +r_{u,K_s}^{1,1}+3r_{u,K_s}^{1,2}\right) . \end{aligned}$$
(5.24)
$$\begin{aligned} c_{v,K_s}^{0,0}= & {} \frac{3}{16}\left( 3r_{v,K_s}^{0,0}+r_{v,K_s}^{1,0} +5r_{v,K_s}^{2,0}\right) ,\, \end{aligned}$$
(5.25)
$$\begin{aligned} c_{v,K_s}^{1,0}= & {} \frac{3}{16}\left( r_{v,K_s}^{0,0}+3r_{v,K_s}^{1,0} +5r_{v,K_s}^{2,0}\right) ,\, \end{aligned}$$
(5.26)
$$\begin{aligned} c_{v,K_s}^{2,0}= & {} \frac{15}{16}\left( r_{v,K_s}^{0,0}+r_{v,K_s}^{1,0} +3r_{v,K_s}^{2,0}\right) ,\, \end{aligned}$$
(5.27)
$$\begin{aligned} a_{v,K_s}^{0,1}= & {} \frac{9}{16}\left( 3r_{v,K_s}^{0,1}+r_{v,K_s}^{1,1} +5r_{v,K_s}^{2,1}\right) ,\, \end{aligned}$$
(5.28)
$$\begin{aligned} c_{v,K_s}^{1,1}= & {} \frac{9}{16}\left( r_{v,K_s}^{0,1} +3r_{v,K_s}^{1,1}+5r_{v,K_s}^{2,1}\right) ,\, \end{aligned}$$
(5.29)
$$\begin{aligned} c_{v,K_s}^{2,1}= & {} \frac{45}{16}\left( r_{v,K_s}^{0,1} +r_{v,K_s}^{1,1}+3r_{v,K_s}^{2,1}\right) . \end{aligned}$$
(5.30)
Then, we have
$$\begin{aligned} \varvec{R}[\Pi _{K_i}\varvec{E}](O)= & {} \left[ \begin{array}{ccc} \frac{1}{sh}\left( c_{u,K_s}^{0,0}+c_{u,K_s}^{0,1} -c_{u,K_s}^{0,2} \right) \\ \frac{1}{sh} \left( c_{v,K_s}^{0,0}+c_{v,K_s}^{1,0}-c_{v,K_s}^{2,0}\right) \end{array}\right] \nonumber \\= & {} \left[ \begin{array}{ccc} \frac{-3}{16sh}\left( r_{u,K_s}^{0,0}+r_{u,K_s}^{0,1}+5r_{u,K_s}^{0,2}\right) \\ \frac{-3}{16sh}\left( r_{v,K_s}^{0,0}+r_{v,K_s}^{1,0}+5r_{v,K_s}^{2,0}\right) \end{array} \right] \nonumber \\=: & {} \left[ \begin{array}{llll} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_1 \\ (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_2 \end{array} \right] . \end{aligned}$$
(5.31)
From (5.18) by careful calculation, we list the right hands in (5.31) as follows:
$$\begin{aligned} r_{u,K_s}^{0,0}= & {} 2\left[ \left( a_{u,K_{1}}^{0,0}+a_{u,K_{2}}^{0,0}\right) \frac{5}{3}s^{2} +\left( a_{u,K_{1}}^{1,0}-a_{u,K_{2}}^{1,0}\right) \frac{5}{3}\left( 1-s\right) s^{2} +\left( a_{u,K_{1}}^{0,1}+a_{u,K_{2}}^{0,1}\right) \nonumber \right. \\&\times \left( 3-\frac{5}{3}s\right) s +\left( a_{u,K_{1}}^{1,1}-a_{u,K_{2}}^{1,1}\right) \left( 1-s\right) \left( 3-\frac{5}{3}s\right) s +\left( a_{u,K_{1}}^{0,2}+a_{u,K_{2}}^{0,2}\right) \nonumber \\&\left. \times \left( \frac{7}{3}s^{3}-\frac{10}{3}s^{2}\right) +\left( a_{u,K_{1}}^{1,2}-a_{u,K_{2}}^{1,2}\right) \left( s-s^{2}\right) \left( \frac{7}{3}s^{2}-\frac{10}{3}s\right) \right] + 2 \left[ \left( a_{u,K_{3}}^{0,0} \nonumber \right. \right. \\&\left. +a_{u,K_{4}}^{0,0}\right) \left( 1-\frac{1}{3}s\right) s +\left( a_{u,K_{3}}^{1,0}-a_{u,K_{4}}^{1,0}\right) \left( s-s^{2}\right) \left( 1-\frac{1}{3}s\right) +\left( a_{u,K_{3}}^{0,1} \nonumber \right. \\&\left. +a_{u,K_{4}}^{0,1}\right) \frac{1}{3}s^{2} +\left( a_{u,K_{3}}^{1,1}-a_{u,K_{4}}^{1,1}\right) \left( s-s^{2}\right) \frac{1}{3}s +\left( a_{u,K_{3}}^{0,2}+a_{u,K_{4}}^{0,2}\right) \nonumber \\&\left. \times \frac{1}{3}s^{2}\left( s-2\right) +\left( a_{u,K_{3}}^{1,2}-a_{u,K_{4}}^{1,2}\right) \left( s-s^{2}\right) \left( \frac{1}{3}s^{2}-\frac{2}{3}s\right) \right] , \end{aligned}$$
(5.32)
$$\begin{aligned} r_{u,K_s}^{0,1}= & {} 2\left[ \big (a_{u,K_{1}}^{0,0}+a_{u,K_{2}}^{0,0}\big ) \frac{1}{3}s^{2} + \big ( a_{u,K_{1}}^{1,0}-a_{u,K_{2}^{1,0}} \big )\big (s-s^{2}\big )\frac{1}{3}s \nonumber \right. \\&+\big (a_{u,K_{1}}^{0,1}+a_{u,K_{2}}^{0,1}\big )\big (1-\frac{1}{3}s\big )s +\big (a_{u,K_{1}}^{1,1}-a_{u,K_{2}}^{1,1}\big )\big (s-s^{2}\big )\big (1-\frac{1}{3}s\big ) \nonumber \\&\left. +\big (a_{u,K_{1}}^{0,2}+a_{u,K_{2}}^{0,2}\big )\frac{1}{3}s^{2}\big (s-2\big ) +\big (a_{u,K_{1}}^{1,2}-a_{u,K_{2}}^{1,2}\big )\big (s-s^{2}\big )\frac{1}{3}s\big (s-2\big ) \right] \nonumber \\&+ 2\left[ \big (a_{u,K_{3}}^{0,0}+a_{u,K_{4}}^{0,0}\big )\big (3-\frac{5}{3}s\big )s +\big (a_{u,K_{3}}^{1,0}-a_{u,K_{4}}^{1,0}\big )\big (s-s^{2}\big )\big (3-\frac{5}{3}s\big ) \nonumber \right. \\&+\big (a_{u,K_{3}}^{0,1}+a_{u,K_{4}}^{0,1}\big )\frac{5}{3}s^{2} +\big (a_{u,K_{3}}^{1,1}-a_{u,K_{4}}^{1,1}\big )\big (s-s^{2}\big )\frac{5}{3}s +\big (a_{u,K_{3}}^{0,2} \nonumber \\&\left. +a_{u,K_{4}}^{0,2}\big ) \big (\frac{7}{3}s^{3}-\frac{10}{3}s^{2}\big ) +\big (a_{u,K_{3}}^{1,2}-a_{u,K_{4}}^{1,2}\big )\big (s-s^{2}\big ) \big (\frac{7}{3}s^{2}-\frac{10}{3}s\big ) \right] , \end{aligned}$$
(5.33)
$$\begin{aligned} r_{u,K_s}^{0,2}= & {} 2\left[ \left( a_{u,K_{1}}^{0,0}+a_{u,K_{2}}^{0,0}\right) \frac{-1}{2}s^{2} +\left( a_{u,K_{1}}^{1,0}-a_{u,K_{2}}^{1,0}\right) \left( s-s^{2}\right) \frac{-1}{2}s \nonumber \right. \\&+\left( a_{u,K_{1}}^{0,1}+a_{u,K_{2}}^{0,1}\right) \left( \frac{-4}{3}s+\frac{1}{2}s^{2}\right) +\left( a_{u,K_{1}}^{1,1}-a_{u,K_{2}}^{1,1}\right) \left( s-s^{2}\right) \nonumber \\&\times \left( \frac{-4}{3}+\frac{1}{2}s\right) +\left( a_{u,K_{1}}^{0,2}+a_{u,K_{2}}^{0,2}\right) \left( \frac{-8}{15}s^{3}+s^{2}\right) +\left( a_{u,K_{1}}^{1,2}-a_{u,K_{2}}^{1,2}\right) \nonumber \\&\left. \times \left( s-s^{2}\right) \left( \frac{-8}{15}s^{2}+s\right) \right] + 2\left[ \left( a_{u,K_{3}}^{0,0}+a_{u,K_{4}}^{0,0}\right) \left( \frac{-4}{3}s+\frac{1}{2}s^{2}\right) \nonumber \right. \\&+\left( a_{u,K_{3}}^{1,0}-a_{u,K_{4}}^{1,0}\right) \left( s-s^{2}\right) \left( \frac{-4}{3}+\frac{1}{2}s\right) +\left( a_{u,K_{3}}^{0,1}+a_{u,K_{4}}^{0,1}\right) \frac{-1}{2}s^{2} \nonumber \\&+\left( a_{u,K_{3}}^{1,1}-a_{u,K_{4}}^{1,1}\right) \left( s-s^{2}\right) \frac{-1}{2}s +\left( a_{u,K_{3}}^{0,2}+a_{u,K_{4}}^{0,2}\right) \left( \frac{-8}{15}s^{3}+s^{2}\right) \nonumber \\&\left. +\left( a_{u,K_{3}}^{1,2}-a_{u,K_{4}}^{1,2}\right) \left( s-s^{2}\right) \left( \frac{-8}{15}s^{2}+s\right) \right] , \end{aligned}$$
(5.34)
$$\begin{aligned} r_{v,K_s}^{0,0}= & {} 2\left[ \left( a_{v,K_{1}}^{0,0}+a_{v,K_{3}}^{0,0}\right) \frac{5}{3}s^{2} +\left( a_{v,K_{1}}^{0,1}-a_{v,K_{3}}^{0,1}\right) \left( s-s^2\right) \frac{5}{3}s +\left( a_{v,K_{1}}^{1,0} +a_{v,K_{3}}^{1,0}\right) \nonumber \right. \\&\times \left( 3s-\frac{5}{3}s^2\right) +\left( a_{v,K_{1}}^{1,1}-a_{v,K_{3}}^{1,1}\right) \left( s-s^2\right) \left( 3-\frac{5}{3}s\right) +\left( a_{v,K_{1}}^{2,0}+a_{v,K_{3}}^{2,0}\right) \nonumber \\&\times \left. \left( \frac{7}{3}s^{3}-\frac{10}{3}s^{2}\right) +\left( a_{v,K_{1}}^{2,1}-a_{v,K_{3}}^{2,1}\right) \left( s-s^{2}\right) \left( \frac{7}{3}s^{2}-\frac{10}{3}s\right) \right] \nonumber \\&+ 2 \left[ \left( a_{v,K_{2}}^{0,0}+a_{v,K_{4}}^{0,0}\right) \left( s-\frac{1}{3}s^2\right) +\left( a_{v,K_{2}}^{0,1}-a_{v,K_{4}}^{0,1}\right) \left( s-s^{2}\right) \left( 1-\frac{1}{3}s\right) \nonumber \right. \\&+\left( a_{v,K_{2}}^{1,0}+a_{v,K_{4}}^{1,0}\right) \frac{1}{3}s^{2} +\left( a_{v,K_{2}}^{1,1}-a_{v,K_{4}}^{1,1}\right) \left( s-s^{2}\right) \frac{1}{3}s +\left( a_{v,K_{2}}^{2,0} \nonumber \right. \\&\left. \left. +a_{v,K_{4}}^{2,0}\right) \left( \frac{1}{3}s^{3}-\frac{2}{3}s^2\right) +\left( a_{v,K_{2}}^{2,1}-a_{v,K_{4}}^{2,1}\right) \left( s-s^{2}\right) \left( \frac{1}{3}s^{2}-\frac{2}{3}s\right) \right] , \end{aligned}$$
(5.35)
$$\begin{aligned} r_{v,K_s}^{1,0}= & {} 2\big [ \big (a_{v,K_{1}}^{0,0}+a_{v,K_{3}}^{0,0}\big )\frac{1}{3}s^{2} + \big ( a_{v,K_{1}}^{0,1}-a_{v,K_{3}}^{0,1} \big )*\big (s-s^{2}\big )\frac{1}{3}s \nonumber \\&+\big (a_{v,K_{1}}^{1,0}+a_{v,K_{3}}^{1,0}\big )\big (1-\frac{1}{3}s\big )s +\big (a_{v,K_{1}}^{1,1}-a_{v,K_{3}}^{1,1}\big )\big (s-s^{2}\big )\big (1-\frac{1}{3}s\big ) \nonumber \\&+\big (a_{v,K_{1}}^{2,0}+a_{v,K_{3}}^{2,0}\big )\frac{1}{3}s^{2}\big (s-2\big ) +\big (a_{v,K_{1}}^{2,1}-a_{v,K_{3}}^{2,1}\big )\big (s-s^{2}\big )\frac{1}{3}s\big (s-2\big ) \big ] \nonumber \\&+ 2\big [ \big (a_{v,K_{2}}^{0,0}+a_{v,K_{4}}^{0,0}\big )\big (3-\frac{5}{3}s\big )s +\big (a_{v,K_{2}}^{0,1}-a_{v,K_{4}}^{0,1}\big )\big (s-s^{2}\big )\big (3-\frac{5}{3}s\big ) \nonumber \\&+\big (a_{v,K_{2}}^{1,0}+a_{v,K_{4}}^{1,0}\big )\frac{5}{3}s^{2} +\big (a_{v,K_{2}}^{1,1}-a_{v,K_{4}}^{1,1}\big )\big (s-s^{2}\big )\frac{5}{3}s +\big (a_{v,K_{2}}^{2,0} \nonumber \\&+a_{v,K_{3}}^{2,0}\big ) \big (\frac{7}{3}s^{3}-\frac{10}{3}s^{2}\big ) +\big (a_{v,K_{2}}^{2,1}-a_{v,K_{4}}^{2,1}\big ) \big (s-s^{2}\big )\big (\frac{7}{3}s^{2}-\frac{10}{3}s\big ) \big ], \end{aligned}$$
(5.36)
$$\begin{aligned} r_{v,K_s}^{2,0}= & {} 2\left[ \left( a_{v,K_{1}}^{0,0}+a_{v,K_{3}}^{0,0}\right) \frac{-1}{2}s^{2} +\left( a_{v,K_{1}}^{0,1}-a_{v,K_{3}}^{0,1}\right) \left( s-s^{2}\right) *\frac{-1}{2}s \nonumber \right. \\&+\left( a_{v,K_{1}}^{1,0}+a_{v,K_{3}}^{1,0}\right) \left( \frac{-4}{3}s+\frac{1}{2}s^{2}\right) +\left( a_{v,K_{1}}^{1,1}-a_{v,K_{3}}^{1,1}\right) \left( s-s^{2}\right) \left( \frac{-4}{3}+\frac{1}{2}s\right) \nonumber \\&\left. +\left( a_{v,K_{1}}^{2,0}+a_{v,K_{3}}^{2,0}\right) \left( \frac{-8}{15}s^{3}+s^{2}\right) +\left( a_{v,K_{1}}^{2,1}-a_{v,K_{3}}^{2,1}\right) \left( s-s^{2}\right) \left( \frac{-8}{15}s^{2}+s\right) \right] \nonumber \\&+ 2\left[ \left( a_{v,K_{2}}^{0,0}+a_{v,K_{4}}^{0,0}\right) \left( \frac{-4}{3}s+\frac{1}{2}s^{2}\right) +\left( a_{v,K_{2}}^{0,1}-a_{v,K_{4}}^{0,1}\right) \left( s-s^{2}\right) \left( \frac{-4}{3}+\frac{1}{2}s\right) \nonumber \right. \\&+\left( a_{v,K_{2}}^{1,0}+a_{v,K_{4}}^{1,0}\right) \frac{-1}{2}s^{2} +\left( a_{v,K_{2}}^{1,1}-a_{v,K_{4}}^{1,1}\right) \left( s-s^{2}\right) \frac{-1}{2}s +\left( a_{v,K_{2}}^{2,0} \nonumber \right. \\&\left. \left. +a_{v,K_{4}}^{2,0}\right) \left( \frac{-8}{15}s^{3}+s^{2}\right) +\left( a_{v,K_{2}}^{2,1}-a_{v,K_{4}}^{2,1}\right) \left( s-s^{2}\right) \left( \frac{-8}{15}s^{2}+s\right) \right] . \end{aligned}$$
(5.37)
From (5.32)–(5.34) and Lemma 3.4, The first component in (5.31) is as follows:
$$\begin{aligned} \big (\varvec{R}[\Pi _{K_i}\varvec{E}]\big (O\big )\big )_1= & {} \frac{-3}{16sh}\big (r_{u,K_s}^{0,0}+r_{u,K_s}^{0,1}+5r_{u,K_s}^{0,2}\big ) \nonumber \\= & {} \frac{-3\times 2}{16sh}\big \{ \big (a_{u,K_{1}}^{0,0}+a_{u,K_{2}}^{0,0}\big )s^{2} +\big (a_{u,K_{1}}^{1,0}-a_{u,K_{2}}^{1,0}\big )\big (1-s\big )s^{2} \nonumber \\&+\big (a_{u,K_{1}}^{0,1}+a_{u,K_{2}}^{0,1}\big )\big (\frac{-8}{3}s+2s^2\big ) +\big (a_{u,K_{1}}^{1,1}-a_{u,K_{2}}^{1,1}\big )\big (s-s^{2}\big ) \nonumber \\&\times \big (\frac{-8}{3}+2s\big ) +\big (\frac{-3}{4h}\int _{K_1}E_1dK +\frac{-3}{4h}\int _{K_2}E_1dK \big ) s^2 \nonumber \\&+\big (\frac{-9}{4h}\int _{K_1}E_1X_{K_1}dK -\frac{-9}{4h}\int _{K_2}E_1X_{K_2}dK\big ) \big (s-s^{2}\big )s \big \} \nonumber \\&+ \frac{-3\times 2}{16sh} \big \{ \big (a_{u,K_{3}}^{0,0}+a_{u,K_{4}}^{0,0}\big )\big (\frac{-8}{3}s+2s^{2}\big ) +\big (a_{u,K_{3}}^{1,0}-a_{u,K_{4}}^{1,0}\big ) \nonumber \\&\times \big (s-s^{2}\big ) \big (\frac{-8}{3}+2s\big ) +\big (a_{u,K_{3}}^{0,1}+a_{u,K_{4}}^{0,1}\big )s^{2} +\big (a_{u,K_{3}}^{1,1}-a_{u,K_{4}}^{1,1}\big ) \nonumber \\&\times \big (s-s^{2}\big )s +\big (\frac{-3}{4h}\int _{K_3}E_1dK +\frac{-3}{4h}\int _{K_4}E_1dK\big )s^{2} \nonumber \\&+\big (\frac{-9}{4h}\int _{K_3}E_1X_{K_3}dK -\frac{-9}{4h}\int _{K_4}E_1X_{K_4}dK\big )\big (s-s^{2}\big )s \big \}. \end{aligned}$$
(5.38)
From Lemma 3.4 and the third order Taylor expansion of \(E_1\) at O:
$$\begin{aligned} E_1=E_1(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}(x\frac{\partial }{\partial x} +y\frac{\partial }{\partial y})^{n}E_1(O) +O(h^{4})\Vert \partial ^4E_{1}\Vert _{\infty }, \end{aligned}$$
we obtain
$$\begin{aligned} a_{u,K_1}^{0,0}= & {} \frac{1}{4}\int _{-h}^{0} \left\{ E_1(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}[x\frac{\partial }{\partial x} +(-h)\frac{\partial }{\partial y}]^{n}E_1(O) \right\} dx \nonumber \\&+O(h^{5})\Vert \partial ^4E_{1}\Vert _{\infty }, \end{aligned}$$
(5.39)
$$\begin{aligned} a_{u,K_1}^{0,1}= & {} \frac{1}{4}\int _{-h}^{0} \left[ E_1(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}(x\frac{\partial }{\partial x} +0\times \frac{\partial }{\partial y})^{n} E_1(O) \right] dx \nonumber \\&+O(h^{5})\Vert \partial ^4u_{1}\Vert _{\infty }, \end{aligned}$$
(5.40)
$$\begin{aligned} a_{u,K_1}^{1,0}= & {} \frac{3}{4}\int _{-h}^{0} \left\{ E_1(O)+\sum \limits _{n=1}^{2} \frac{1}{n!}[x\frac{\partial }{\partial x} +(-h)\frac{\partial }{\partial y}]^{n}E_1(O) \right\} X_{K_1} dx \nonumber \\&+O(h^{5})\Vert \partial ^4E_1\Vert _{\infty }, \end{aligned}$$
(5.41)
$$\begin{aligned} a_{u,K_1}^{1,1}= & {} \frac{3}{4}\int _{-h}^{0} \left\{ E_1(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}[x\frac{\partial }{\partial x} +0\times \frac{\partial }{\partial y}]^{n}E_1(O) \right\} X_{K_1}dx \nonumber \\&+O(h^{5})\Vert \partial ^4E_{1}\Vert _{\infty }, \end{aligned}$$
(5.42)
$$\begin{aligned} \int _{K_1}E_1dxdy= & {} \int _{K_1} \left[ E_{1}(O) +\sum _{n=1}^{3}\frac{1}{n!} ( x\frac{\partial }{\partial _{x}} +y\frac{\partial }{\partial _{y}} )^{n}E_{1}(O) \right] dK \nonumber \\&+O(h^{5})\Vert \partial ^{4}E_{1}\Vert _{\infty }, \end{aligned}$$
(5.43)
$$\begin{aligned} \int _{K_1}E_1X_{K_1}dxdy= & {} \int _{K_1} \left[ E_{1}(O) +\sum _{n=1}^{3}\frac{1}{n!} (x\frac{\partial }{\partial _{x}} +y\frac{\partial }{\partial _{y}}) ^{n}E_{1}(O) \right] \times X_{K_1}dK \nonumber \\&+O(h^{5})\Vert \partial ^4E_{1}\Vert _{\infty }. \end{aligned}$$
(5.44)
Similarly, we obtain \(a_{u,K_i}^{0,0}\), \(a_{u,K_i}^{0,1}\), \(a_{u,K_i}^{1,0}\), \(a_{u,K_i}^{1,1}\), \(\int _{K_i}u_1dxdy\), \(\int _{K_i}u_1X_{K_i}dxdy\, (\hbox {i}=2,3,4)\). Substituting (5.39)–(5.44) into (5.38), we obtain
$$\begin{aligned} (\varvec{R}[\Pi _{K_i}\varvec{E}][O])_1= & {} \frac{-3}{16sh}\left( r_{u,K_s}^{0,0}+r_{u,K_s}^{0,1}+5r_{u,K_s}^{0,2}\right) \\= & {} E_{1}(O)+\frac{-3}{8}\left( \frac{2}{9}-\frac{2s}{3}\right) h^2 \frac{\partial ^2 E_1}{\partial x^2}(O)+O(h^4)\Vert \partial ^4E_1\Vert _{\infty }. \end{aligned}$$
By choosing \(s=\frac{1}{3}\), we obtain the first component with fourth order accuracy.
Next, let us deal with the second component. From (5.35)–(5.37) and Lemma 3.4, the second component \((\varvec{R}[\Pi _{K_i}\varvec{u}][O])_2\) is as follows:
$$\begin{aligned} \left( \varvec{R}[\Pi _{K_i}\varvec{E}][O]\right) _2= & {} \frac{-3}{16sh}\left( r_{v,K_s}^{0,0}+r_{v,K_s}^{1,0}+5r_{v,K_s}^{2,0}\right) \nonumber \\= & {} \frac{-3}{16sh}\times 2\left[ \left( a_{v,K_{1}}^{0,0}+a_{v,K_{3}}^{0,0}\right) s^{2} +\left( a_{v,K_{1}}^{0,1}-a_{v,K_{3}}^{0,1}\right) \left( s-s^2\right) s \nonumber \right. \\&+\left( a_{v,K_{1}}^{1,0}+a_{v,K_{3}}^{1,0}\right) \left( -\frac{8}{3}s+2s^2\right) +\left( a_{v,K_{1}}^{1,1}-a_{v,K_{3}}^{1,1}\right) \left( s-s^2\right) \nonumber \\&\times \left( \frac{-8}{3}+2s\right) +\left( \frac{-3}{4h}\int _{K_1}E_2dK +\frac{-3}{4h}\int _{K_3}E_2dK \right) s^{2} \nonumber \\&\left. +\left( \frac{-9}{4h}\int _{K_1}E_2Y_{K_1}dK -\frac{-9}{4h}\int _{K_3}E_2Y_{K_3}dK \right) \left( s-s^{2}\right) s \right] \nonumber \\&+ \frac{-3}{16sh}*2 \left[ \left( a_{v,K_{2}}^{0,0}+a_{v,K_{4}}^{0,0}\right) \left( -\frac{8}{3}s+2s^2\right) +\left( a_{v,K_{2}}^{0,1}-a_{v,K_{4}}^{0,1}\right) \nonumber \right. \\&\times \left( s-s^{2}\right) \left( -\frac{8}{3}+2s\right) +\left( a_{v,K_{2}}^{1,0}+a_{v,K_{4}}^{1,0}\right) s^{2} +\left( a_{v,K_{2}}^{1,1}-a_{v,K_{4}}^{1,1}\right) \nonumber \\&\times \left( s-s^{2}\right) s +\left( \frac{-3}{4h}\int _{K_2}E_2dK +\frac{-3}{4h}\int _{K_4}E_2dK \right) s^2 \nonumber \\&\left. +\left( \frac{-9}{4h}\int _{K_2}E_2Y_{K_2}dK - \frac{-9}{4h}\int _{K_4}E_2Y_{K_4}dK \right) \left( s-s^{2}\right) s \right] . \end{aligned}$$
(5.45)
From Lemma 3.4 and the third order Taylor expansion of \(E_2\) at O:
$$\begin{aligned} E_2=E_2(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}(x\frac{\partial }{\partial x} +y\frac{\partial }{\partial y})^{n}E_2(O)+O(h^{4})\Vert \partial ^4E_{2}\Vert _{\infty }, \end{aligned}$$
we deduce
$$\begin{aligned} a_{v,K_1}^{0,0}= & {} \frac{1}{4}\int _{-h}^{0} \left\{ E_2(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}[(-h)\frac{\partial }{\partial x} +y\frac{\partial }{\partial y}]^{n}E_2(O) \right\} dy \nonumber \\&+O(h^{5})\Vert \partial ^4u_{1}\Vert _{\infty }, \end{aligned}$$
(5.46)
$$\begin{aligned} a_{v,K_1}^{1,0}= & {} \frac{1}{4}\int _{-h}^{0} \left\{ E_2(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}[0\times \frac{\partial }{\partial x} +y\frac{\partial }{\partial y}]^{n}E_2(O) \right\} dy \nonumber \\&+O(h^{5})\Vert \partial ^4E_{2}\Vert _{\infty }, \end{aligned}$$
(5.47)
$$\begin{aligned} a_{v,K_1}^{0,1}= & {} \frac{3}{4}\int _{-h}^{0} \left\{ E_2(O)+\sum \limits _{n=1}^{3} \frac{1}{n!}[(-h)\frac{\partial }{\partial x} +y\frac{\partial }{\partial y}]^{n}E_2(O) \right\} Y_{K_1} dy \nonumber \\&+O(h^{5})\Vert \partial ^4E_{2}\Vert _{\infty }, \end{aligned}$$
(5.48)
$$\begin{aligned} a_{v,K_1}^{1,1}= & {} \frac{3}{4}\int _{-h}^{0} \left[ E_2(O)+\sum \limits _{n=1}^{2} \frac{1}{n!}(0\times \frac{\partial }{\partial x} +y\frac{\partial }{\partial y})^{n}E_2(O) \right] Y_{K_1}ds \nonumber \\&+O(h^{5})\Vert \partial ^4E_{2}\Vert _{\infty }, \end{aligned}$$
(5.49)
$$\begin{aligned} \int _{K_1}E_2dK= & {} \int _{K_1}\left[ E_2(O)+\sum \limits _{n=1}^{2} \frac{1}{n!}\left( x\frac{\partial }{\partial x} +y\frac{\partial }{\partial y}\right) ^nE_2(O)\right] dK \nonumber \\&+O(h^5)\Vert \partial ^4 E_2\Vert _{\infty }, \end{aligned}$$
(5.50)
$$\begin{aligned} \int _{K_1}E_2Y_{K_2}dK= & {} \int _{K_1}\left[ E_2(O)+\sum \limits _{n=1}^{2} \frac{1}{n!}\left( x\frac{\partial }{\partial x} +y\frac{\partial }{\partial y}\right) ^nE_2(O)\right] Y_{K_2}dK \nonumber \\&+O(h^5)\Vert \partial ^4 E_2\Vert _{\infty }. \end{aligned}$$
(5.51)
Similarly, we obtain \(a_{v,K_i}^{0,0}\), \(a_{v,K_i}^{0,1}\), \(a_{v,K_i}^{1,0}\), \(a_{v,K_i}^{1,1}\), \(\int _{K_i}E_2dK\), \(\int _{K_i}E_2Y_{K_i}dK\,(\hbox {i}=2,3,4)\).
Substituting (5.46)–(5.51) into (5.45), we obtain
$$\begin{aligned} (\varvec{R}[\Pi _{K_i}\varvec{E}](O))_2= & {} \frac{-3}{16sh}\left( r_{v,K_s}^{0,0}+r_{v,K_s}^{1,0}+5r_{v,K_s}^{2,0}\right) \\= & {} E_{2}(O)+\frac{-3}{8}\left( \frac{2}{9}-\frac{2s}{3}\right) h^2 \frac{\partial ^2 E_2}{\partial y^2}(O)+O(h^4)\Vert \partial ^4E_2\Vert _{\infty }. \end{aligned}$$
By choosing \(s=\frac{1}{3}\), we can also obtain fourth order accuracy for the second component.
