## 4 Error Estimates of Iterative Schemes for the Galerkin Spectral Method
K^{\mathrm{T}} \tag{25}\]  
where \(R_{r\times r}=dig\{\sigma_{1},\sigma_{2},\cdots,\sigma_{r}\}\), and r = rank(Z), \(\sigma_{k}(k=1,2,\cdots,r)\) is the positive singular value of matrix \(Z\) in descending order, which is arranged in descending order (\(\delta_{r}\geq\delta_{r-1}\geq\cdots\geq\delta_{2}\geq\delta_{1}\geq 0\)). \(U=(\psi_{1},\psi_{2},\cdots,\psi_{G})\) is an orthogonal matrix of order G \(\times\) G, \(\psi_{k}(k=1,2,\cdots,G)\) is the standard orthogonal eigenvector of matrix \(ZZ^{T}\), which is arranged in the order of \(\sigma_{k}\), \(K=(\varphi_{1},\varphi_{2},\cdots,\varphi_{L})\) is an orthogonal matrix of order \(L\times L\), \(\varphi_{k}(k=1,2,\cdots,L)\) is the standard orthogonal eigenvector of matrix \(Z^{T}Z\), and its arrangement order is the same as \(\psi_{k}\). Normally, the number of numerical solution components L of the selected [PERSON] spectral iteration scheme is smaller than the number of spatial grid nodes G. It can be seen that the order of matrix \(Z^{T}Z\) is smaller than that of \(ZZ^{T}\), but \(Z^{T}Z\) and \(ZZ^{T}\) have the same non-negative eigenvalues. If necessary, \(\lambda_{k}=\delta_{k}^{2}\) can be renumbered in a descending order to meet the following requirements: \(\lambda_{r}\geq\lambda_{r-1}\geq\cdots\geq\lambda_{2}\geq\lambda_{1}\geq 0\). Here, we first calculate the non-zero eigenvalues and eigenvectors of matrix \(Z^{T}Z\), secondly, we use the above relationship to establish Equation (26):  
\[\psi_{k}=\frac{1}{\sqrt{\lambda_{k}}}Z\varphi_{k},\;k=1,2,\cdots,r \tag{26}\]  
It can be found that the eigenvector of matrix \(Z^{T}Z\) is \(\psi_{k}(1\leq k\leq r\leq L)\), and finally we obtain matrix \(U\) and matrix \(K\); here, we assume \(\psi=(\psi_{1},\psi_{2},\cdots,\psi_{c})\), and select the first \(e\) positive singular values \((\sigma_{1},\sigma_{2},\cdots,\sigma_{e})\), of matrix \(R_{r\times r}\) to construct a diagonal matrix, \(R_{e\times e}=dig\{\sigma_{1},\sigma_{2},\cdots,\sigma_{e}\}(e\leq r)\), and we can obtain (27):  
\[Z_{e}=U \begin{bmatrix}R_{e\times e}&0_{e\times(L-e)}\\ 0_{(G-e)\times e}&0_{(G-e)\times(L-e)}\end{bmatrix} K^{\mathrm{T}} \tag{27}\]so as to obtain the relationship \(Z_{e}=\psi\psi^{T}Z\). Third step: Generation of the POD base. We specify that the norm of matrix \(Z\) is \(\left\|Z\right\|_{2,2}=\sup_{u\in R^{L}}\left\|Zu\right\|_{2}/\left\|Z\right\|_{2}\), (here \(\left\|u\right\|_{2}\) is the norm of vector \(u\)). According to the relationship between matrix norm and spectral radius, we know that (28) from [29] is:  
\[\left\|Z-Z_{e}\right\|_{2,2}=\left\|Z-\psi\psi^{T}Z\right\|_{2,2}=\sqrt{ \lambda_{d+1}} \tag{28}\]  
It can be seen that the [PERSON] spectral method based on the POD reduced-order extrapolation method should meet the following conditions (29):  
\[\left\{ \begin{array}{l}\sum\limits_{i=1}^{L}\left\|u_{H}^{i}-\sum\limits_{k= 1}^{e}\left(u_{H^{i}}^{i},\psi^{k}\right)\psi^{k}\right\|^{2}\rightarrow\min \\ \psi\psi^{T}=I\end{array} \right. \tag{29}\]  
The minimum value of the optimization result of Equation (29) means that the basis of the Galerkin spectral method based on the POD algorithm is the maximum projection of the original system on the POD basis.