## 7 Error Estimation of POD Reduced-Order Extrapolation Galerkin Spectral Method
\[\left|u_{H_{k}}^{i}-u_{\alpha_{k}}^{i}\right|\leq E(i)\sqrt{\lambda_{e+1}} \tag{45}\]  
where \(1\leq k=n+1+m(H+1)\leq G=(H+1)^{2}\partial\leq i\leq I\), we can get:  
\[E(i)= \begin{cases}1&1\leq i\leq L\\ 3\exp[(\frac{2\Delta H\left\|C\right\|_{2}}{\rho c}+\frac{2\Delta H\left\|D \right\|_{2}}{\rho c})(i-\text{L}+1)]&L+1\leq i\leq I\end{cases}  \tag{46}\]  
It can be seen that the precise solution of the transient nonlinear heat conduction Equation (1) at point \((x_{n},y_{m},i\Delta t)\) and the numerical solution of the POD reduced dimension extrapolation Galerkin spectral method iteration scheme (34) have the following error estimates:  
\[\left|u(x_{n},y_{m},\Delta t)-u_{\varepsilon_{k}}^{i}\right|=O(E(i)\sqrt{ \lambda_{e+1}},\Delta t,H^{-T}) \tag{47}\]  
where \(,1\leq k=n+1+m(H+1)\leq G=(H+1)^{2},0\leq n,m\leq H,2\leq T\leq H\). **Note 2:** Looking at the whole process of the iterative format of the POD reduced-order extrapolation Galerkin spectral method, it can be found that the error factor \(\sqrt{\lambda_{e+1}}\) comes from the optimization harvest of the Galerkin spectral method iterative format after dimensionality reduction. In particular, when \(i=L,L+1,\cdots,I\), the extrapolation process produces \(3\exp[\left(\frac{2\Delta H\left\|C\right\|_{2}}{\rho c}+\frac{2\Delta H \left\|D\right\|_{2}}{\rho c}\right)(i-L+1)]\) factor, it can be said that whether it is necessary to reconstruct the temperature field and replace the POD base completely depends on the variation range of the above two factors. Furthermore, whether the selection of POD reduced-order extrapolation method basis is appropriate requires the following two conditions. 1. In order to ensure that the POD reduced-order extrapolation method has a small error in the Galerkin spectral method iteration process, the \"optimal\" POD basis must be selected, that is, meeting the conditions \(\sqrt{\lambda_{e+1}}\leq sup\left\{H^{-T},\Delta t\right\}\). 2. If \(3\sqrt{\lambda_{e+1}}\ exp[\left(\frac{2\Delta H\left\|C\right\|_{2}}{\rho c}+ \frac{2\Delta H\left\|D\right\|_{2}}{\rho c}\right)(i-L+1)]>sup\left\{H^{-T}, \Delta t\right\}\), we need to obtain a new POD base, if there is a factor \(\sqrt{\lambda_{e+1}}\) that makes \(\sqrt{\lambda_{e+1}}\ exp[\left(\frac{2\Delta t\left\|C\right\|_{2}+2\Delta H \left\|D\right\|_{2}}{\rho c}\right)\] \((i-L+1)]=O\left\{H^{-T},\Delta t\right\}(L+1\leq i\leq I)\) true, this shows that the POD reduced-order iteration is convergent, and the original POD basis remains. Algorithm Implementation of Iterative Scheme of Reduced-Order Extrapolation Galerkin Spectral Method  
The implementation of the algorithm of iterative formats (32) and (33) of the reduced-order extrapolation Galerkin spectral method can be completed in the following five steps [31; 32; 33]. Step 1: We can select spatial nodes as Chebyshev-Gauss-Lobatto type configuration points and construct a Galerkin spectral iteration format reasonably based on the initial time step \(\Delta t\leq 4.8H^{-3}\), then, calculating the initial L-step solution vector \(u_{H}^{i}=\left(u_{H_{1}}^{i},u_{H_{2}}^{i},\cdots,u_{H_{G}}^{i}\right)^{T}\), \(\left(i=1,2,\cdots,L\right)\), and definition: \(u_{H_{k}}^{i}=u_{H}^{i}(x_{n},y_{m}),(0\leq n,\)\(m\leq H,i=1,2,\cdots,L,G=\left(H+1\right)^{2})\). Step 2: We can build the sample matrix \(\ Z=\left(u_{H_{k}}^{i}\right)_{G\times L}\), and calculate the eigenvalues \(\lambda_{i}(i=1,2,\cdots,L)\) and eigenvectors \(\varphi_{i}(i=1,2,\cdots,L)\) of matrix \(Z^{T}Z\). Among them, the feature values are arranged in a descending order, thusly: \((\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{i}\geq\lambda_{i+1}=\cdots= \lambda_{L}=0)\). Step 3: Define condition \(\sqrt{\lambda_{e+1}}\leq\sup\left\{H^{-T},\Delta t\right\}\) and determine POD basis \(\psi=\left(\psi_{1},\psi_{2},\cdots,\psi_{e}\right)\) by known precision \(\sup\left\{H^{-T},\Delta t\right\}\), and \(\psi_{k}\) is solved by \(\frac{1}{\sqrt{\lambda_{k}}}Z\varphi_{k}\).