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

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.

Condition number and preprocess of the finite element equation of two point boundary value problems with 2-degree Lagrange shape function

For the Ill posed 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.