## 5 Conclusions  
Traditionally, the preconditioning of solution algorithms for linear systems of equations discretized using the finite element method has relied on the conventional ILU factorization method. This approach often results in unsatisfactory preconditioning effects, accompanied by increased computational costs and challenges in achieving parallelism. To address these issues, we propose a preconditioning algorithm named G-Pre, which is based on graph-theoretic methods, to enhance the efficiency of solving linear systems of equations. In the proposed method, a combination of graph partitioning, graph reordering, and ILU factorization was employed to concentrate the non-zero elements of the coefficient matrix, thereby improving data access efficiency and facilitating parallelism through matrix partitioning. Numerical experiments were conducted using test matrices, and an actual finite element model was utilized for further testing. The results demonstrated that the G-Pre preconditioning algorithm proposed in this paper can improve the condition number optimization by 2.6-fold compared with that of ILU preconditioning; the solution efficiency \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Original** & **ILU** & **G-Pre** \\ \hline Model 1 & 0.0692 & 0.0227 & 0.0396 \\ Model 2 & 0.327 & 0.242 & 0.155 \\ Model 3 & 15.2 & 4.56 & 1.45 \\ Model 4 & 64.8 & 49.6 & 28.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of solution times for finite element simulation models. Figure 9: A visualization of the different models. can be improved 2.1- and 4.3-fold compared with ILU preconditioning and direct solvers. In terms of iterative convergence performance, the number of iterations can be reduced by an average of 30.8% compared with ILU preconditioning. Meanwhile, the resultant error can be maintained at 1%. It has been proven that the G-Pre preconditioning algorithm can improve the convergence performance of the solver and shorten the solution time. However, the current G-Pre preconditioning algorithm still has some shortcomings. First, there is no clear optimization effect of the algorithm in the small-scale matrix solving process, and in some situations, it is not as good as ILU preconditioning. Secondly, the initial solution of the algorithm needs to be further optimized so that the initial residuals are too large, which affects the iterative convergence process and the error of the results. Finally, the possibility of combining more solution algorithms and preconditioning algorithms should be considered to further broaden the applicability of the G-Pre preconditioning algorithm. In future work, further refinement is planned to address the shortcomings of the current algorithm in order to improve its performance. First, the performance of the G-Pre preconditioning algorithm in small-scale matrices needs to be improved, and the introduction of an adaptive algorithm can be considered to reduce the time cost of the graph algorithm in small-scale matrices. In addition, the setting of the initial solution needs to be further optimized to reduce the impact on iterative convergence. Meanwhile, a supervised optimization procedure can also be considered to judge the quality of the initial solution and to reset it if it does not meet the requirements. Finally, introducing more advanced techniques such as a graph neural network in the preconditioning algorithm can also further improve its efficiency. Further research will be carried out in future work to make continuous efforts to realize an efficient G-Pre preconditioning algorithm. Conceptualization: M.C., J.L. and [PERSON]; methodology: [PERSON]; software: [PERSON] and [PERSON]; validation: [PERSON]; formal analysis: [PERSON] and [PERSON]; investigation: [PERSON]; resources: [PERSON]; writing--original draft preparation: [PERSON] and [PERSON]; writing--review and editing: [PERSON], [PERSON] and [PERSON]; visualization: [PERSON]; supervision: [PERSON], [PERSON] and [PERSON].