一种用于计算矢量有限元方程组的不完全分解预处理方法

An incomplete factorization preconditioning method for solving vector finite element equations

提出一种不完全分解预处理方法,并结合迭代法计算矢量有限元方程组。预处理方法采用基于拓展乔里斯基分解的多波前法对有限元方程组的系数矩阵进行分解和更新,并采用基本线性代数系统库函数计算稠密矩阵乘来保证算法内层循环的高效率。该预处理算法在对系数矩阵进行数值分解前引入缩放矩阵以改善矩阵条件数。针对有限元方程组系数矩阵稀疏或部分稀疏的特性,提出一种新的舍弃策略以保证不完全分解的精度和提高预条件子的构造时间。通过与直接法对比,从时间花费与内存占用两方面,分析了该算法的计算性能。理论和数值实验表明,提出的预处理方法能大大减少计算时间与分解过程所占用的内存,同时保证了计算的准确性和有效性。

In this paper, an incomplete decomposition preconditioning method is presented combining with iterative method to solve vector finite element equations. In the preconditioning process, the coefficient matrix is decomposed and updated by the Multifrontal algorithm based on the Expanded Cholesky method. The basic linear algebra subprogram is used to cal- culate multiplication of dense matrices to ensure the computational efficiency of inner loop. Unlike the traditional precondi- tioning technique, the incomplete decomposition method borrows the concept of scaling matrix to improve the conditioning of the coefficient matrix. Moreover, a new version of dropping scheme is proposed to make the preconditioning method robust. The presented method and direct method are compared in terms of time and memory. Theory and numerical experiments show that the presented preconditioning method can greatly reduce the time and memory of decomposition, and ensure the accuracy and validity of the calculation.