两点边值问题2次Lagrange形函数有限元方程的条件数和预处理
Condition number and preprocess of the finite element equation of two point boundary value problems with 2-degree Lagrange shape function

求解积分形式的两点边值问题时，基于2次Lagrange形函数形成的有限元方程是病态正定对称五对角方程组. 为了寻找该方程的病态原因，提出根据系数矩阵的特别结构，设计出预条件子的方法，并将产生病态的因子定义为致病因子，预条件子称为去病因子. 分析结果表明，使用去病因子进行预处理，可以保证系数矩阵的正定对称性，迭代求解时，预条件子几乎不增加迭代的计算量，预处理后的条件数接近1.
For the ill conditioned problem of the positive definite symmetric five diagonal finite element systems formed solving two-point boundary value problems of integral form using finite element method based on 2-degree Lagrange shape function. The method for looking up the reason producing ill called pathogenic factor and designing preconditioner called eliminate ill factor to eliminating the ill-condition was obtained based on the special structure of the systems. The results of the analysis shows that pretreatment using the eliminate ill factor，can guarantee positive definite symmetry of the coefficient matrix， and the condition number is close to 1 after pretreatment without causing more computing