## 3 Numerical Example
### Example 1
Therefore, the use of ADOF-FEM can reduce the total cost of calculating the transient temperature field, and the more elements, the more cost is saved. The computational time of Refined1 with adaptive DOFs is 82.26% of that without adaptive DOFs, while the computational time of Refined2 with adaptive DOFs is only 61.03% of that without adaptive DOFs. Therefore, the more elements there are, the higher the computational cost, which can be saved by using ADOF-FEM. In addition to discussing the advantages of ADOF-FEM in saving computational cost, it is also necessary to analyze the accuracy of the algorithm, that is, whether there is a large \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Time(ms)** & **Coarse** & \multicolumn{2}{c}{**Refined1**} & \multicolumn{2}{c}{**Refined2**} \\  & **FEM** & **ADOF-FEM** & **FEM** & **ADOF-FEM** \\ \hline Assemble global conductivity matrix & 561 & 1308 & 3081 & 5059 & 9673 \\ Set DOFs & 1 & 2 & 59 & 5 & 539 \\ Matrix factorization & 4171 & 12,364 & 8374 & 57,686 & 26,867 \\ Solve equation & 823 & 2460 & 1758 & 4164 & 3756 \\ Total & 5556 & 16,134 & 13,272 & 66,914 & 40,835 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The computational time of Example 1 (Unit ms). loss of accuracy leading to the distortion of the calculation results. In this paper, the root mean square error (RMSE) between the FEM result and the analytical solution is used to represent the accuracy in the computational time domain. The RMSE of the FEM result with or without adaptive DOFs is shown in Table 2, where the master DOFs at positions (0,0), (50,0), and (100,0) are unchanged and the master DOFs at positions (25,0), (75,0) are transformed into the slave DOFs when using ADOF-FEM. As can be seen from Table 2, the calculation accuracy of the finite element increases with the increase of the number of elements, which is consistent with the general law. The RMSE results of refined meshes with adaptive DOFs are slightly different from those of the original refined meshes; in other words, when using ADOF-FEM there is only a small loss of the accuracy, and even better accuracy at few positions due to a coincidence of calculation conditions. Combined with Figure 3, it can be seen that the number of DOFs of the refined meshes gradually decreases with time, which is close to the number of DOFs of the coarse mesh eventually, but the accuracy of the refined mesh with adaptive DOFs is much higher than that of the coarse mesh. In addition, the loss of the accuracy is also different at different positions. The accuracy on the surface can be consistent with that of the refined mesh since the DOFs on the surface are not condensed, while the loss of the accuracy at position (50,0) is greater than that of other positions, which is due to a large number of DOFs near this position condensed at the same time. Moreover, the improvement in accuracy at position (75,0) may be due \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Position**} & **Coarse** & \multicolumn{3}{c}{**Refined1**} & \multicolumn{3}{c}{**Refined2**} \\  & **FEM** & **FEM** & **ADOF-FEM** & **Relative** & **FEM** & **ADOF-FEM** & **Relative** \\  & **RMSE** & **RMSE** & **RMSE** & **Difference** & **RMSE** & **RMSE** & **Difference** \\ \hline (0,0) & 0.8067 & 0.3886 & 0.3904 & 0.45\% & 0.2850 & 0.2861 & 0.35\% \\ (25,0) & 0.7898 & 0.3600 & 0.3601 & 0.01\% & 0.2663 & 0.2666 & 0.15\% \\ (50,0) & 0.5742 & 0.2929 & 0.2951 & 0.77\% & 0.2258 & 0.2271 & 0.59\% \\ (75,0) & 0.9603 & 0.2731 & 0.2720 & -0.39\% & 0.2368 & 0.2366 & -0.06\% \\ (100,0) & 3.0676 & 1.3934 & 1.3935 & 0.01\% & 0.8351 & 0.8353 & 0.03\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The RMSE of the result of Example 1. Figure 3: The change of the number of DOFs in Example 1.  
to the fact that its relative error after the transformation of the master DOF to the slave DOF is independent of the cumulative error in the previous time step caused by the time discretization of that DOF.