两点边值问题3次Lagrange形函数有限元方程的条件数和预处理

大型病态稀疏线性方程组的求解是科学计算和工程应用中的重要问题之一,采用预处理方法,通过降低条件数来减少病态是解决这一问题的关键。基于3次Lagrange形函数,用有限元方法将积分形式两点边值问题的求解转化成病态七对角方程组的求解。通过研究该方程组的特殊结构,分析了该方程的条件数,找到产生病态的因子(致病因子)。将系数矩阵的大范数部分分解成几个简单矩阵的特殊组合,基于这种特殊分解,设计出预条件子(去病因子),并对预条件子的性能进行了定量分析。结果表明,该预条件子的使用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 preprocessing of the finite element equation of two point boundary value problems with cubic Lagrange shape function

Solving large sparse ill-conditioned linear equations is very important in scientific computing and engineering applications.The key to solve the problem is reducing the condition number by preprocessing.The finite element system formed in solving two-point boundary value problems of integral form using the finite element method based on cubic Lagrange shape functions is converted into a system of ill-conditioned seven diagonal equations,and the condition number of the system was analyzed by studying the special structure of the equation,and the factor causing ill-conditioning was found.The big norm part of the coefficient matrix was decomposed into an assemble of several simple matrices.The preconditioner was obtained based on the decomposition,and performance analysis of the preconditioner was given in a quantitative manner.The results of analysis show that the condition number is close to 1 after pretreatment without causing more computation.