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：佩斯金提出的沉浸边界法（J. Comput. Phys., 1977）适用于流体-结构相互作用或复杂几何形状问题。该方法利用独立于几何结构的规则笛卡尔网格，形成一种稳健的数值格式，可高效处理沉浸式可变形结构。此外，沉浸边界法已推广至预定义运动问题及其他含给定边界条件的偏微分方程。传统上，沉浸边界法针对此类问题采用罚函数力来近似满足边界条件，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的方程来寻找未知力，该操作可能需要大量Krylov方法迭代。由于时变问题需在每个时间步进行求解，若无预处理，该方法可能效率低得难以实施。本文提出一种新型良态沉浸边界法边界问题公式——沉浸边界双层（IBDL）方法。我们通过泊松问题和亥姆霍兹问题的应用实例，展示该方法相对于原始约束方法的高效性。在该双层公式中，未知边界分布的方程对应良态第二类积分方程，仅需少量Krylov方法迭代即可高效求解，且迭代次数与网格尺寸和浸没边界点间距无关。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公式还可处理诺伊曼边界条件。