A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems
A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems
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DOI：
10.1016/j.jcp.2023.111924
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发表时间：
2023-01-19
影响因子：
4.1
通讯作者：
Zhao，Shan
中科院分区：
文献类型：
--
作者：
Ren，Yiming;Zhao，Shan
本文提出了一种四阶增广匹配界面边界（AMIB）方法，用于求解长方体区域内光滑材料界面的三维椭圆界面问题。在长方体域的边界上，四阶AMIB方法可以处理不同类型的边界条件，包括虚拟值生成中的Dirichlet、Neumann、Robin及其混合组合。此外，引入了零填充解，使得快速傅里叶变换（FFT）算法在边界附近仍然有效。在处理内部接口时，提出了一种四阶光线投射匹配接口和边界（MIB）方案，该方案强制沿法线方向的跳跃条件以计算虚拟值。与现有的MIB方案相比，光线投射方案自然地绕过了拐角问题，在处理复杂几何图形时变得更加鲁棒。基于在界面和边界附近产生的虚拟值，通过引入笛卡尔导数跳变作为辅助变量，可以在包括角点在内的各种不规则点上校正四阶中心差。这就产生了一个扩大的线性系统，该系统可以通过Schur补过程和离散拉普拉斯算子的FFT反演有效地求解。我们进行了大量的数值实验，以测试所提出的光线铸造AMIB方法在拐角处理中的数值精度、效率和鲁棒性。数值结果表明，对于n× n× n均匀网格，光线投射AMIB方案不仅在处理解和解梯度的各种界面和边界时保持了四阶精度，而且总体效率达到了O （n3log (n)）阶。
In this paper, a fourth order augmented matched interface and boundary (AMIB) method is proposed for solving a three-dimensional elliptic interface problem which involves a smooth material interface inside a cuboid domain. On the boundary of the cuboid domain, the fourth order AMIB method can handle different types of boundary conditions, including Dirichlet, Neumann, Robin and their mix combinations in fictitious value generation. Moreover, zero-padding solutions are introduced so that the fast Fourier transform (FFT) algorithm is still valid near the boundary. In dealing with the interior interface, a fourth order ray-casting matched interface and boundary (MIB) scheme is proposed, which enforces the jump conditions along the normal direction for calculating fictitious values. Comparing with the existing MIB scheme, the ray-casting scheme naturally bypasses the corner issue and becomes more robust in handling complex geometry. Based on fictitious values generated near interface and boundary, the fourth order central difference can be corrected at various irregular points including corner points, by introducing Cartesian derivative jumps as auxiliary variables. This gives rise to an enlarged linear system, which can be efficiently solved by the Schur complement procedure together with the FFT inversion of the discrete Laplacian. Extensive numerical experiments have been carried to test the proposed ray-casting AMIB method for numerical accuracy, efficiency, and robustness in corner treatment. The numerical results demonstrate that the ray-casting AMIB scheme not only maintains a fourth order of accuracy in treating various interfaces and boundaries for both solutions and solution gradients, but also attains an overall efficiency on the order of O (n 3 log n) for a n× n× n uniform grid.