A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains
A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains
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DOI：
10.1016/j.jcp.2021.110762
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发表时间：
2022-01
期刊：
影响因子：
--
通讯作者：
Y. Ren;Hongsong Feng;Shan Zhao
中科院分区：
文献类型：
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作者：
Y. Ren;Hongsong Feng;Shan Zhao
对于含有非规则区域和Robin边界条件的椭圆型边值问题，目前还没有一种数值方法能达到四阶收敛和O(N logN)的效率，其中N代表系统的总自由度。基于匹配界面和边界(MIB)格式和快速傅立叶变换(FFT)格式，提出了一种新的求解此类问题的有限差分方法。首先，提出了一种光线投射MIB格式来处理不同类型的边界条件，包括Dirichlet、Neumann、Robin以及它们的混合组合。光线投射MIB格式通过将不规则区域封闭在足够大的立方体区域中，沿边界法线方向施加边界条件，在不规则区域外产生必要的虚值，从而形成拉普拉斯算子的高阶中心差分离散。其次，建立了一种扩展的MIB公式，其中笛卡尔导数跳跃被重构为边界上的辅助变量。通过将这些变量视为未知量，在增广系统的Schur补解中，利用FFT算法可以有效地求出离散拉普拉斯算子。通过对二维和三维不同椭圆边界值问题的数值计算，验证了增广MIB方法的精度和效率。数值结果表明，新算法在处理不规则区域和复杂边界条件时，不仅达到了四阶精度，而且保持了FFT的效率。
For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O (N log N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.