橢圓界面問題的藕合界面方法

Abstract

在這篇論文中提出了一個可以在任意維度、複雜界面、卡式坐標的網格上解決橢圓界面問題的方法：藕合界面方法(Coupling Interface Method)。它允許左端項的係數上、右邊的來源項甚至是方程的解上面在跨過界面時都可以是不連續的。而在精準度上，它提供了一個一階方法(CIM1)、二階方法(CIM2)及兩者混合(CIM)的方法。 在一維的時候，一階方法(CIM1)是由界面兩邊各自的線性逼近所導出的。而延伸到高維度的時候是一個可以從各個維度處理的方法。 它需要透過在每一個維度上的界面條件建立聯立方程組去連結各個維度上的一階微分的資訊。最後所導出來的差分格式在二維的時候是一個五點差分，而三維的時候是一個七點差分。它是一個非常緊密的差分格式。類似地，在一維時的二階方法(CIM2)是由面兩邊各自的二階逼近所導出的。而延伸至高維度時，在各個維度上透過界面條件的聯立方程組去連結每一個維度上的二階微分的資訊。所有的二階主要偏導數($u_{x_k x_k}$)都可以從聯立方程組求出。而交叉微分項($u_{x_j x_k}$)則是由單邊差分所逼近。這是減少用來做插值的所需要的點數的關鍵。最後所導出來的差分格式在二維的時候必須用到8個點，而三維的時候則需要12-14個點。 在使用的限制上，一階方法(CIM1)需要限制界面與每一條網格線段的時候只能有一個交點，這個限制是非常寬鬆的，而且只要網格夠密的時候都可以辦得到。而二階方法(CIM2)則需要一個類似的條件，即界面與中央點在每一個維度的兩邊的網格線段上只能有一個交點，但是在大多數情形下，網格夠密的時候這個條件也會成立。因此，在實作上我們將每一個網格點分成三類：內部點、正常的界面點以及例外的點。在內部點上我們會用標準的中央差分法，而正常的界面點上我們會用二階方法，而在剩下的例外點上我們會用一階方法。這個混合的差分方法我們稱為藕合界面方法(Coupling Interface Method)。在大多數的情況下它會是一個二階方法。這是因為通常在 d 維度的時候，界面大多是一個 d-1 維的曲面。所以正常的界面點有大約有 d-1 維度的量。但例外的點數大多是常數。 數值上在此篇論文中會去討論一維情況二階方法(CIM2)所形成矩陣的特徵值，它都是正的實數。而且它Condition number的行為跟Laplacian是非常類似的。另外，只要所產生網格的夠密的時候，我們會證明一個充份條件，使得該方法所產生的聯立方程組一定是可解的。在求解此差分方法所形成的大型稀疏矩陣的時候我們是利用代數多重網格法來求解。而在收斂性測試上，我們發現藕合界面方法所產生的誤差比其他目前常用或是已發表的二階方法來得少。而且，藕合界面方法通過了許多複雜界面的測試。因此，我們有信心它是一個在處理複雜界面時相當具競爭力的方法。

We propose a coupling interface method (CIM) under Cartesian grid for solving elliptic complex interface problems in arbitrary dimensions, where the coefficients, the source terms, and the solutions may be discontinuous or singular across the interfaces. It consists of a first-order version (CIM1) and a second-order version (CIM2). In one dimension, the CIM1 is derived from a linear approximation on both sides of the interface. The method is extended to high dimensions through a dimension-by-dimension approach. To connect information from each dimension, a coupled equation for the first-order derivatives is derived through the jump conditions in each coordinate direction. The resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions. Similarly, the CIM2 is derived from a quadratic approximation in each dimension. In high dimensions, a coupled equation for the principal second-order derivatives $u_{x_k x_k}$ is derived through the jump conditions in each coordinate direction. The cross derivatives are approximated by one-side interpolation. This approach reduces the number of grid points needed for one-side interpolation. The resulting stencil involves 8 grid points in two dimensions and 12-14 grid points in three dimensions. A numerical study for the condition number of the resulting linear system of the CIM2 in one dimension has been performed. It is shown that the condition number has the same behavior as that of the discrete Laplacian, independent of the relative location of the interface in a grid cell. Further, we also give a proof of the solvability of the coupling equations, provided the curvature $kappa$ of the interface satisfies $kappa hle Const.$, where $h$ is the mesh size. The CIM1 requires that the interface intersects each grid segment (the segment connecting two adjacent grid points) at most once. This is a very mild restriction and is always achievable by refining meshes. The CIM2 requires basically that the interface does not intersect two adjacent grid segments simultaneously. In practice, we classify the underlying Cartesian grid points into interiors, normal on-fronts, and exceptionals, where a standard central finite difference method, the CIM2, and the CIM1 are adopted, respectively. This hybrid CIM maintains second order accuracy in most applications due to the fact that usually in $d$ dimensions, the number of normal on-front grid points is $O(h^{1-d})$ and the number of the exceptional points is $O(1)$. Numerical convergence tests for the CIM1 and CIM2 are performed. A comparison study with other interface methods is also reported. Algebraic multigrid method is employed to solve the resulting linear system. Numerical tests demonstrate that CIM1 and CIM2 are respectively first order and second order in the maximal norm with less error as compared with other methods. In addition, this hybrid CIM passes many tests of complex interface problems in two and three dimensions. Therefore, we believe that it is a competitive method for complex interface problems.