A fast iterative algorithm for elliptic interface problems
A fast iterative algorithm for elliptic interface problems
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DOI：
10.1137/s0036142995291329
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发表时间：
1998-02-01
影响因子：
2.9
通讯作者：
Li, ZL
中科院分区：
文献类型：
--
作者：
Li, ZL
本文提出一种求解椭圆型方程的快速二阶精度迭代法。（beta（x，y）del u）= f（x，y）在二维空间中的矩形区域Ω中。我们假设有一个不规则的界面，系数β，解u及其导数，和/或源项f可能有跳跃。我们特别感兴趣的情况下，系数β是分段常数和跳跃β是大的。界面可以与底层笛卡尔网格对齐，也可以不对齐。在我们的方法中的想法是预处理的微分方程之前，应用浸没接口方法提出的LeVeque和李[SIAM J. Numer.分析：4（1994），pp. 1019-1044]。为了利用快速泊松求解器在矩形区域上，中间未知函数，跨越接口的正常导数的跳跃，被引入。我们的离散化是相当于使用相应的泊松方程在该地区的二阶差分格式，和一个二阶离散的Neumann接口条件。因此，二阶精度得到保证。一个GMRES迭代求解Schur补系统从离散。本文还提出了一种新的加权最小二乘方法来近似网格函数的界面量。数值实验和分析。在解决舒尔补系统的迭代次数似乎是独立的系数和网格大小的跳跃。
A fast, second-order accurate iterative method is proposed for the elliptic equationdel .(beta(x,y)del u) = f(x,y)in a rectangular region Omega in two-space dimensions. We assume that there is an irregular interface across which the coefficient beta, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [SIAM J. Numer. Anal., 4 (1994), pp. 1019-1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.