【作者】 徐森林；

【导师】 舒适；

【作者基本信息】 湘潭大学 ， 计算数学， 2011， 硕士

【摘要】 有限体元(FVE)法是一种常用的偏微分方程离散化方法最近发展起来的二阶混合有限体元法由于具有保持局部守恒性以及控制体结构简单等特点，受到人们的关注，本文将为该方法对应的离散系统设计快速算法首先，针对‘种含跳系数的椭圆问题在分层基下的二阶混台有限体元离散系统，利用基于两种常用预条件子，ILu和AMG的PGMRES(m)法进行求解数值结果表明，它们的求解效率都不高，其迭代次数依赖于网格规模和跳系数因此，有必要发展新的高效预条件了接着，奉文为上述离散系统设计了两种预条件了：块对角预条件了和两水平预条件了对于前者，在‘致三角形剖分下给出了相应的理论分析，得到了预条件系统谱条件数‘致有界的结论数值结果表明，我们设计的预条件了是高效的，相应PGMRES(m)法的迭代次数明显减少，它个依赖于网格规模，且对跳系数和重启参数m的选取均个敏感更多还原

【Abstract】 Finite Volume Element (FVE) method is one of the discretization methods for partialdifferential equations. Second order mixed-type finite volume element method which wasdeveloped recently has many advantages, such as being able to preserve local conservationof certain physical quantities. Fast algorithms for the corresponding discretization systemswill be discussed in this paper.Firstly, ILU- and AMG-GMRES(m) are employed to solve the linear systems arisingfrom second order mixed-type finite volume element scheme for an elliptic problem withjump coefficients. Numerical results show that the numbers of iteration of the two PGM-RES (m) are unstable which strongly depend on not only the mesh size but also the jumpcoefficient. Therefore, it is quite necessary to develop newly efficient preconditioners.Secondly, two new preconditioners are designed for the above linear systems, i.e., blockdiagonal and two-level preconditioner. For consistent mesh, theoretical analysis of the for-mer is given, and it is proved that the spectral condition number of the preconditioned sys-tem is uniformly bounded. Numerical results show that our preconditioners are stable andefficient. The numbers of iteration of the corresponding PGMRES (m) are significantly re-duced, furthermore, they are independent of mesh size and insensitive to jump coefficient orrestart parameter.更多还原