The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.

翻译：Peskin提出的浸没边界法（J. Comput. Phys., 1977）对于涉及流固耦合或复杂几何形状的问题具有重要价值。通过使用独立于几何结构的规则笛卡尔网格，浸没边界框架能够构建稳健的数值格式，有效处理可变形浸没结构。此外，该框架已被推广至受迫运动问题及其他带有给定边界数据的偏微分方程问题。传统上，处理此类问题的浸没边界方法采用惩罚力仅近似满足边界条件，或将其表述为约束问题。后者需通过求解对应病态第一类积分方程来确定未知力。这一过程可能需要大量Krylov子空间迭代，而时变问题在每个时间步均需执行此求解，因而若无预处理，该方法效率极低。本文提出一种新型良态浸没边界格式——浸没边界双层法。我们以该方法在泊松方程和亥姆霍兹方程中的应用为例，展示其相较于原始约束方法的效率优势。在此双层格式中，未知边界分布方程对应于良态第二类积分方程，可通过少量Krylov子空间迭代高效求解，且迭代次数与网格尺寸及浸没边界点间距均无关。该方法在远离边界处收敛，结合局部插值后可在整个偏微分方程域内收敛。此外，原始约束方法仅适用于狄利克雷问题，而IBDL格式还可处理诺伊曼边界条件。