## 4 Error Estimates of Iterative Schemes for the Galerkin Spectral Method  
Using [PERSON]'s series expansion, we get  
\[u_{k}^{i+1}=u_{k}^{i}+\Delta t\frac{\partial u_{k}^{i}}{\partial t}+\frac{ \Delta t^{2}}{2!}\frac{\partial^{2}u_{k}^{i}}{\partial t^{2}}+\frac{\Delta t ^{3}}{3!}\frac{\partial^{3}u_{k}^{i}}{\partial t^{3}}+O\Big{(}\Delta t^{4} \Big{)} \tag{20}\]  
\[u_{k}^{i-1}=u_{k}^{i}-\Delta t\frac{\partial u_{k}^{i}}{\partial t}+\frac{ \Delta t^{2}}{2!}\frac{\partial^{2}u_{k}^{i}}{\partial t^{2}}-\frac{\Delta t ^{3}}{3!}\frac{\partial^{3}u_{k}^{i}}{\partial t^{3}}+O\Big{(}\Delta t^{4} \Big{)} \tag{21}\]  
From (19) and (20), we can get  
\[\frac{u_{k}^{i+1}-u_{k}^{i-1}}{2\Delta t}-\frac{\partial u(x_{n},y_{m},i \Delta t)}{\partial t}=O(\Delta t),k=n+1+m(H+1) \tag{22}\]  
**Theorem 1**. : _The solution component \(u(x_{n},y_{m},i\Delta t)\)of the iterative scheme (19) of the central difference Galerkin spectral method has the following error estimates:_  
\[\left\|u_{H}^{i}(x_{n},y_{m})-u(x,y,i\Delta t)\right\|_{2}=O\Big{(}\Delta t,H ^{-h}\Big{)},2\leq h\leq H+1 \tag{23}\]  
_where \(\left\|\cdot\right\|\) is the usual norm of the vector._ As for the proof of this theorem, which can be given a detailed answer from Equation (22) and reference [25], and we can obtain that its stability condition satisfies \(\Delta t\leq 4.8\) H\({}^{-3}\). Galerkin Spectral Analysis of Two-Dimensional Transient Nonlinear Heat Conduction Equation Based on POD Reduced-Order Extrapolation Structure of POD Base  
The selection of the POD basis is very important when the Galerkin spectral method is used to solve transient nonlinear heat conduction problems based on the POD reduced-order dimension extrapolation algorithm. Obtaining the best POD basis can not only ensure the accuracy of numerical calculation results, but also greatly improve the efficiency of experimental simulation. Therefore, it can be seen that the core problem of the POD reduced-order extrapolation method is to find a set of \"optimal\" orthogonal bases of \(\left\{u^{i}\in L^{2}(\Omega)|i=1,2,\cdots,H\right\}\) on the known function space through an iterative calculation. There is no doubt that the so-called \"optimization\" means that the error generated by the projection of function space onto a finite number of orthogonal bases with relatively low dimensions reaches the minimum. In this way, when using the reduced dimension extrapolation Galerkin spectrum method of the POD method to solve the transient heat conduction problem, it is usually realized through the following steps. The first step: Construct sample matrix. Assume \(u_{k}^{i}=u^{i}(x_{n},y_{m}),u_{H_{k}}^{i}=u_{H}^{i}(x_{n},y_{m}),\)  
Here, \(1\leq k=n+1+m(H+1)\leq G\), \(G=(H+1)^{2}\), \(0\leq n,m\leq H,0\leq i\leq I\), The initial solution \(\left\{u_{H_{k}}^{i}\right\}_{i=1}^{L}(1\leq k\leq G,L<<G)\) of \(L\) (\(L<<I\)) is calculated and recorded as a sample. Thus, the solution of the iterative scheme (19) of the classical central difference Galerkin spectral method can be expressed in sets, and the initial solution A of \(L(L\leq 1)\) group can be calculated, which is recorded as the instantaneous image. Next, we will construct a G \(\times\)\(L\)-order instantaneous image matrix. That is, as sample matrix \(Z\). \[Z= \begin{bmatrix}u_{H_{1}}^{1}&u_{H_{1}}^{2}&\cdots&u_{H_{1}}^{L}\\ u_{H_{2}}^{1}&u_{H_{2}}^{2}&\cdots&u_{H_{2}}^{L}\\ \vdots&\vdots&\ddots&\vdots\\ u_{H_{G}}^{1}&u_{H_{G}}^{2}&\cdots&u_{H_{G}}^{L}\end{bmatrix} _{G\times L} \tag{24}\]  
The second step: SVD decomposition of sample matrix, through SVD decomposition technology (see, e.g., [26; 27; 28]), the sample matrix can be expressed as:  
\[Z=U \begin{bmatrix}R_{r\times r}&0_{r\times(L-r)}\\ 0_{(G-r)\times r}&0_{(G-r)\times(L-r)}\end{bmatrix}