## 2 Methods
### Matrix Preconditioning Algorithm  
To solve large sparse matrices, iterative methods are a better choice. However, the large condition number of large sparse matrices significantly affects the stability, efficiency, and convergence of these iterative methods, which seriously hinders their application in finite element simulation. The convergence characteristics of iterative methods are generally related to the distribution of the eigenvalues of the coefficient matrix of a system of linear equations, especially for Krylov subspace methods. The convergence of these methods is closely related to the density of the eigenvalues of the coefficient matrix; the more concentrated the distribution, the faster the convergence speed [50]. Preconditioning techniques are acceleration methods that are designed to improve the convergence characteristics of iterative methods. These techniques construct a preconditioner \(M\) through the incomplete factorization of matrix \(A\), transforming the original system of linear equations into an equivalent new system. This transformation makes the distribution of the eigenvalues of the coefficient matrix in the new system more concentrated, thereby improving the convergence performance of the iterative solver. Preconditioning techniques are crucial for the successful application of iterative methods in practical problems and have become one of the hot spots in the research of the efficient solving of sparse linear equations [27]. Additionally, the combination of preconditioning techniques and Krylov subspace projection methods is considered to be the best \"general\" iterative solving method [34]. For commonly used iterative algorithms, it is essential to establish a suitable preconditioner to improve the condition number of the coefficient matrix \(A\). This improves the number of iterations required, accelerates convergence, makes the iterative process more stable, and reduces computational cost. Preconditioning technology is the core means to accelerate the convergence of iteration. The most commonly used preconditioning method is ILU factorization. ILU factorization is an enhancement of LU factorization, as illustrated in Figure 1. ILU is a preconditioning method that reduces computational and storage overhead by retaining only some non-zero elements, while decomposing matrix \(A\) into an approximate product of a lower triangular matrix \(L\) and an upper triangular matrix \(U\). This method can effectively mitigate the fill-in phenomenon and reduce memory requirements. Generally, there are two main approaches to preconditioning using incomplete LU factorization. 1. Multilevel incomplete factorization (ILU()): This approach, denoted as ILU(\(l\)), is based on symbolic factorization, which computes the hierarchical level \(l\) of the non-zero elements in the factorization from the non-zero structure of the original matrix \(A\). Elements with a hierarchical level \(l\) greater than a specified threshold are discarded [51]. The hierarchical value of the non-zero elements in the factorization is solely dependent on the distribution of the non-zero elements in \(A\) and is independent of their actual values. If matrix \(A\) is diagonally dominant, the elements at higher levels in the factorization tend to have smaller absolute values during LU factorization. Thus, in incomplete factorization, elements with level values exceeding a given threshold can be discarded [50]. However, for matrices without diagonal dominance, this relationship does not hold, and discarding elements based purely on their hierarchical level may result in unreliable preconditioners due to ignoring the impact of element values [34]. 2. Threshold rounding strategy: Another approach involves using a threshold rounding strategy based on the numerical magnitude of the elements in the factorization [52]. Given a threshold value \(\tau>0\), non-zero elements in the factorization are discarded if their absolute value is less than \(\tau\); otherwise, they are retained.