A Fast Iterative Algorithm for Elliptic Interface Problems

数学 舒尔补语 离散化 分段 泊松方程 广义最小残差法 数学分析 应用数学 线性系统 特征向量 物理 量子力学
作者
Zhilin Li
出处
期刊:SIAM Journal on Numerical Analysis [Society for Industrial and Applied Mathematics]
卷期号:35 (1): 230-254 被引量:247
标识
DOI:10.1137/s0036142995291329
摘要

A fast, second-order accurate iterative method is proposed for the elliptic equation \[ \grad\cdot(\beta(x,y) \grad u) =f(x,y) \] in a rectangularregion $\Omega$ 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.

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