An augmented matched interface and boundary (MIB) method for solving elliptic interface problem
An augmented matched interface and boundary (MIB) method for solving elliptic interface problem
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DOI：
10.1016/j.cam.2019.05.004
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发表时间：
2019-12
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影响因子：
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通讯作者：
Hongsong Feng;Guangqing Long;Shan Zhao
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文献类型：
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作者：
Hongsong Feng;Guangqing Long;Shan Zhao
本文提出了一种二阶精度增广匹配界面和边界（MIB）方法来求解二维逐段常系数椭圆界面问题。增强MIB无缝地结合了标准MIB、增强浸入式接口方法（IIM）和显式跳转IIM的几个关键成分，以产生新的快速接口算法。基于MIB，零阶和一阶跳跃条件在任意弯曲的界面上被强制执行，这在界面附近的笛卡尔节点上产生虚拟值。通过使用这样的虚拟值，提出了一个简单的程序来重建笛卡尔导数跳跃作为辅助变量，并将它们与跳跃校正的泰勒级数展开，这使我们能够恢复的顺序的中心差异在界面上的两个。此外，通过使用Schur补来分离辅助变量和函数值的代数计算，可以通过使用快速傅立叶变换（FFT）来有效地反演离散拉普拉斯算子。数值实验表明，求解辅助方程组的迭代次数与网格尺寸的关系很弱。因此，对于二维n× n维的笛卡尔网格，增广MIB的总计算量约为O（n2logn）.因此，增强的MIB优于经典的MIB在所有情况下，显着减少CPU时间，同时保持相同的二阶精度在处理复杂的接口。
In this paper, a second order accurate augmented matched interface and boundary (MIB) is introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients. The augmented MIB seamlessly combines several key ingredients of the standard MIB, augmented immersed interface method (IIM), and explicit jump IIM, to produce a new fast interface algorithm. Based on the MIB, zeroth and first order jump conditions are enforced across an arbitrarily curved interface, which yields fictitious values on Cartesian nodes near the interface. By using such fictitious values, a simple procedure is proposed to reconstruct Cartesian derivative jumps as auxiliary variables and couple them with the jump-corrected Taylor series expansions, which allow us to restore the order of the central difference across the interface to two. Moreover, by using the Schur complement to disassociate the algebraic computation of auxiliary variables and function values, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). It is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size. As a consequence, the total computational cost of the augmented MIB is about O (n 2 log n) for a Cartesian grid with dimension n× n in 2D. Therefore, the augmented MIB outperforms the classical MIB in all cases by significantly reducing the CPU time, while keeping the same second order of accuracy in dealing with complicated interfaces.