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（J. Comput. Phys., 1977）提出的浸没边界方法适用于涉及流固耦合或复杂几何形状的问题。通过采用与几何形状无关的规则笛卡尔网格，浸没边界框架能够构建鲁棒的数值格式，高效处理可形变浸没结构。此外，该方法已被推广至包含预设运动的问题及其他具有给定边界数据的偏微分方程。传统浸没边界方法处理此类问题时，要么采用仅近似满足边界条件的惩罚力，要么将其构造成约束问题。在后者中，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。这一过程可能需要大量迭代的Krylov子空间方法，而由于时间依赖问题要求在每一步时间步中进行求解，若缺乏预处理，该方法效率极低。本文提出一种新型良态浸没边界公式——浸没边界双层方法，用于处理边值问题。我们通过该方法在泊松与亥姆霍兹问题中的应用，展示其相较于原始约束方法的优越性。在该双层公式中，未知边界分布对应的方程转化为良态第二类积分方程，可通过少量迭代的Krylov方法高效求解，且迭代次数与网格尺寸及浸没边界点间距无关。该方法远离边界处收敛，结合局部插值后可在整个偏微分方程域中收敛。同时，原始约束方法仅适用于狄利克雷问题，而IBDL公式同样支持诺伊曼条件。