Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids) or when the solution is needed in the bulk, nearly-singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case where each object is "simple", ie, its smooth boundary needs only moderately many nodes. We present a kernel-independent method needing only an upsampled smooth surface quadrature, and one dense factorization, for each distinct shape. No (near-)singular quadrature rules are needed. The resulting effective sources are drop-in compatible with fast algorithms, with no local corrections nor bookkeeping. Our extensive numerical tests include 2D FMM-based Helmholtz and Stokes BVPs with up to 1000 objects (281000 unknowns), and a 3D Laplace BVP with 10 ellipsoids separated by $1/30$ of a diameter. We include a rigorous analysis for analytic data in 2D and 3D.
翻译:用于解决椭圆边界值问题的完善的边界整体方法(BVPs)是静态和动态物理模拟的有力工具。当存在许多接近到触摸的边界(例如复杂液体中)或需要大宗溶液时,必须在许多目标中评估近星形综合体。我们表明,从表面密度到有效源代表的线性地图的预先计算使得这项任务效率很高,在每个物体为“质”的常见情况下,其平滑的边界只需要少量多节点。我们提出了一个依靠内核独立的方法,每个不同的形状只需要一个上至触的平滑的表面二次曲线和一个密度系数。不需要任何(近距离)近距离的二次二次曲线规则。由此产生的有效来源与快速算法相容,没有本地校正或书本保存。我们的广泛数字测试包括2DFMM-Hlmoltz和Stoks BVPs,最多有1000个未知对象(281000美元未知的),以及一个3D型直径直径的BP2和1D1美元的直径直径。