求解带有界面和不规则区域的椭圆问题的浸入界面有限元方法
发布时间：2018-09-01 07:10
【摘要】：在许多实际应用中,所研究的对象经常由不同物质组成,不同的物质通过界面相互分开.如果对这些实际问题建立微分方程的数学模型,那么在微分方程中，不仅参数是间断的,而且在界面上也需要满足一些界面条件.在本文中,我们考虑使用非匹配网格的浸入界面有限元方法来求解这些问题.对于界面问题,Li et al. (Numer Math 96:61-98,2003)提出的非协调浸入界面有限元方法在文献中已有许多深入的研究.该方法实际上是通过对传统的P1协调元进行修正,使其在界面单元上满足界面条件.非协调浸入界面有限元方法不仅程序实现简单而且其离散解在L2和H1范数下能达到最优精度.然而由于修正的基函数的不连续性，非协调浸入界面有限元方法在L∞范数下不能达到最优的二阶精度.尽管Li et al. (Numer Math 96:61-98,2003)也构造了完全二阶精度的协调浸入界面有限元,但是其构造比较复杂,不易程序实现.在本文的第一部分,我们通过改进原来的非协调浸入界面有限元方法,提出了一种对称相容的浸入界面有限元方法.该方法依旧使用非协调浸入界面有限元空间,因此不仅保留了非协调浸入界面有限元的一些优点,而且对称相容,更重要的是具有二阶精度.这个方法的思想是,针对修正的基函数的不连续性,在双线性型中加入一些修正项来保持相容性和对称性.接下来,我们把这个对称相容浸入界面有限元方法推广到带非其次跳跃界面条件的界面问题中.在本文的第二部分,我们提出两个增广的浸入界面有限元方法来求解界面问题和不规则区域问题.该方法实际上是一种快速迭代法.增广技巧首先由Li (SIAM J. Numer. Anal.35:230-254,1998)提出,并且应用在有限差分方法中.我们简单地把增广技巧应用到有限元框架中.得到第一个增广浸入界面有限元方法.在增广方法中，通过引进一个或多个在界面或边界上的增广变量,使得我们能更加容易的离散原来的微分方程.增广变量应该选取到使得界而或边界条件得以满足.增广方法成功的关键经常依靠一种插值来把增广变量和原来的微分方程耦合在一起.这通过最小二乘插值(系数不定)来完成.奇异值分解被用来求解插值系数.接下来,借助有限元的性质,我们提出了第二个增广浸入界面有限元方法.在这个新的方法中,我们避免了使用最小二乘插值.因此,这个新的增广方法比原来使用最小二乘插值的增广方法效率更高,程序更简单.然后我们把这个方法推广到带有狄氏边界条件的不规则区域问题中.我们提供了许多数值实验来展示这个新方法的精度和效率,包括带有任意界面/不规则区域的例子和系数跳跃很大的例子.数值结果表明GMRES迭代次数和网格尺度无关,而且和系数跳跃基本无关.
[Abstract]:In many practical applications, the objects studied are often composed of different substances, which are separated from each other through interfaces. If the mathematical model of differential equation is established for these practical problems, in the differential equation, not only the parameters are discontinuous, but also some interface conditions must be satisfied in the interface. In this paper, we consider to solve these problems by using the immersion interface finite element method (FEM) with mismatched meshes. Li et al. for interface problems (Numer Math 96: 61-98 / 2003) the nonconforming immersion interface finite element method has been deeply studied in the literature. In fact, by modifying the traditional P1 coordination element, the method can satisfy the interface condition on the interface unit. The nonconforming immersion interface finite element method is not only simple to be realized but also its discrete solution can achieve the optimal accuracy under L _ 2 and H _ 1 norm. However, due to the discontinuity of the modified basis function, the non-conforming immersion interface finite element method can not achieve the optimal second-order accuracy under L 鈭，
本文编号：2216482
[Abstract]:In many practical applications, the objects studied are often composed of different substances, which are separated from each other through interfaces. If the mathematical model of differential equation is established for these practical problems, in the differential equation, not only the parameters are discontinuous, but also some interface conditions must be satisfied in the interface. In this paper, we consider to solve these problems by using the immersion interface finite element method (FEM) with mismatched meshes. Li et al. for interface problems (Numer Math 96: 61-98 / 2003) the nonconforming immersion interface finite element method has been deeply studied in the literature. In fact, by modifying the traditional P1 coordination element, the method can satisfy the interface condition on the interface unit. The nonconforming immersion interface finite element method is not only simple to be realized but also its discrete solution can achieve the optimal accuracy under L _ 2 and H _ 1 norm. However, due to the discontinuity of the modified basis function, the non-conforming immersion interface finite element method can not achieve the optimal second-order accuracy under L 鈭，
本文编号：2216482
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