求解帶有界面和不規(guī)則區(qū)域的橢圓問題的浸入界面有限元方法
發(fā)布時間：2018-09-01 07:10
【摘要】：在許多實際應用中,所研究的對象經常由不同物質組成,不同的物質通過界面相互分開.如果對這些實際問題建立微分方程的數學模型,那么在微分方程中，不僅參數是間斷的,而且在界面上也需要滿足一些界面條件.在本文中,我們考慮使用非匹配網格的浸入界面有限元方法來求解這些問題.對于界面問題,Li et al. (Numer Math 96:61-98,2003)提出的非協調浸入界面有限元方法在文獻中已有許多深入的研究.該方法實際上是通過對傳統的P1協調元進行修正,使其在界面單元上滿足界面條件.非協調浸入界面有限元方法不僅程序實現簡單而且其離散解在L2和H1范數下能達到最優(yōu)精度.然而由于修正的基函數的不連續(xù)性，非協調浸入界面有限元方法在L∞范數下不能達到最優(yōu)的二階精度.盡管Li et al. (Numer Math 96:61-98,2003)也構造了完全二階精度的協調浸入界面有限元,但是其構造比較復雜,不易程序實現.在本文的第一部分,我們通過改進原來的非協調浸入界面有限元方法,提出了一種對稱相容的浸入界面有限元方法.該方法依舊使用非協調浸入界面有限元空間,因此不僅保留了非協調浸入界面有限元的一些優(yōu)點,而且對稱相容,更重要的是具有二階精度.這個方法的思想是,針對修正的基函數的不連續(xù)性,在雙線性型中加入一些修正項來保持相容性和對稱性.接下來,我們把這個對稱相容浸入界面有限元方法推廣到帶非其次跳躍界面條件的界面問題中.在本文的第二部分,我們提出兩個增廣的浸入界面有限元方法來求解界面問題和不規(guī)則區(qū)域問題.該方法實際上是一種快速迭代法.增廣技巧首先由Li (SIAM J. Numer. Anal.35:230-254,1998)提出,并且應用在有限差分方法中.我們簡單地把增廣技巧應用到有限元框架中.得到第一個增廣浸入界面有限元方法.在增廣方法中，通過引進一個或多個在界面或邊界上的增廣變量,使得我們能更加容易的離散原來的微分方程.增廣變量應該選取到使得界而或邊界條件得以滿足.增廣方法成功的關鍵經常依靠一種插值來把增廣變量和原來的微分方程耦合在一起.這通過最小二乘插值(系數不定)來完成.奇異值分解被用來求解插值系數.接下來,借助有限元的性質,我們提出了第二個增廣浸入界面有限元方法.在這個新的方法中,我們避免了使用最小二乘插值.因此,這個新的增廣方法比原來使用最小二乘插值的增廣方法效率更高,程序更簡單.然后我們把這個方法推廣到帶有狄氏邊界條件的不規(guī)則區(qū)域問題中.我們提供了許多數值實驗來展示這個新方法的精度和效率,包括帶有任意界面/不規(guī)則區(qū)域的例子和系數跳躍很大的例子.數值結果表明GMRES迭代次數和網格尺度無關,而且和系數跳躍基本無關.
[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，
本文編號：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，
本文編號：2216482
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