We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $\mathcal O(n)$ time, where $n$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nystr\"om quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies, and provide rigorous justification for compression of submatrices via proxy surfaces.
翻译:我们利用最近推出的循环强力骨质化计划的延伸,为复杂表面三个维度的边界整体方程式展示了一个快速直接求解器。对于不高度混凝土的问题,我们的算法对稠密系统矩阵计算了一个类似于${LU}的等级因子化,允许在$\mathcal O(n) 美元的时间里应用反向的O(n)美元,其中美元是表层上未知数字的数量。系数化本身也以系统大小为线性尺度,尽管是一个较大的常数。这个方案建立在一个有层次限制的适应性奥氏族数据结构上,因此与高度不统一的离散化不兼容。此外,这个方案与高层次的精确本地校正 Nystr\\'om 方形方法相配合,将单一的和微弱的绿色功能整合到整体表层中。我们的方法直接适用于计算物理学中的各种问题。我们在这里集中研究它在低至中层频率的声散(由Helmholtz方方方形方程式管理)的性表现。