## 1 Introduction  
The finite element method (FEM) is a powerful numerical computation technique that is utilized to address complex engineering and physical problems. At its core, the FEM involves discretizing the continuous solution domain into a finite number of elements, establishing approximate solutions within these elements, and employing mathematical tools such as the variational principle or weighted residual method to derive an overall approximate solution for the entire domain. The advancement of electronic computers has facilitated the rapid development of the FEM into a sophisticated computational approach, leading to the emergence of a substantial market for finite element analysis software. In finite element analysis (FEA), many scientific computational projects and practical engineering applications often boil down to solving a system of linear equations with one or more large sparse matrices as the coefficient matrices. The physical properties of the problem and the numerical discretization method determine whether these matrices are symmetric. Symmetric matrices typically arise in problems with symmetric physical properties, such as stiffness matrices in elasticity mechanics, where material-eigen-structure relations are usually invertible and symmetric. Asymmetric matrices generally originate from fluid dynamics problems [1], e.g., convection-dominated problems (CDPs) [2]. When significant convective effects are present (e.g., a high-velocity fluid flow), the discretization of convective terms leads to asymmetry. Additionally, nonlinear Navier-Stokes convective terms can disrupt matrix symmetry. Matrix solution problems are now an essential component of finite element simulation and analysis across various fields, including the automotive [3], aerospace [4], construction [5], semiconductor [6], materials [7], agriculture and forestry [8], and medicine [9] industries. Therefore, efficiently solving the system of linear equations discretized using the finite element method remains a focal point of research in FEA. At present, many research institutions and scholars, both at home and abroad, have invested a lot of resources in carrying out in-depth research on algorithms for solving systems of linear equations [10; 11; 12; 13; 14; 15]. The solution algorithms are mainly categorized into two main groups--direct methods [16; 17] and iterative methods [18]. The direct method is a numerical method for finding the exact solution of a system of linear equations via finite-step arithmetic operations. However, with the gradual increase in the size of the solving matrix, the conventional direct method is now no longer applicable, and the iterative method is usually used [19]. For cases where the coefficient matrices are symmetric, the conjugate gradient (CG) method is typically applied. This well-known iterative method is effective for solving large sparse symmetric matrix problems [20]. In contrast, when dealing with asymmetric coefficient matrices, the solution approach differs from that used for symmetric matrices. The commonly adopted algorithm in this scenario is the generalized minimal residual (GMRES) method, which is an iterative technique that is specifically designed for solving large sparse asymmetric linear systems [21; 22]. However, the convergence behavior of the iterative algorithm depends heavily on the size of the condition number of the correlation matrix [23]. To enhance the stability and speed of the iterative convergence process, matrix preconditioning techniques are often applied [24; 25]. This technique transforms the initial system of linear equations by constructing a preprocessor M, which is equivalent to the initial system of linear equations; then, the resulting new system of linear equations for computation is solved [26]. This conversion process aims to make the distribution of the eigenvalues of the coefficient matrix of the new system of equations more centralized, which, in turn, speeds up the convergence rate.