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（J. Comput. Phys., 1977）提出的浸没边界（IB）方法适用于涉及流固耦合或复杂几何形状的问题。该方法利用独立于几何结构的规则笛卡尔网格，构建了能够有效处理浸没可变形结构的稳健数值格式。此外，IB方法已被推广至具有预设运动的问题及其他带给定边界数据的偏微分方程（PDE）问题。针对这类问题的传统IB方法要么引入仅近似满足边界条件的惩罚力，要么将其表述为约束问题。在后一种方法中，需通过求解对应于病态第一类积分方程的方程来寻找未知力。此类操作可能需大量迭代Krylov子空间方法，而由于随时间变化的问题需要在每个时间步执行该求解过程，若无预处理，该方法可能因效率过低而无法应用。本文提出一种新型良态边界值问题的IB格式——浸没边界双层（IBDL）方法。我们展示了该方法在泊松方程与亥姆霍兹问题中的应用，以证明其相较于原始约束方法的效率优势。在该双层格式中，未知边界分布的方程对应于良态第二类积分方程，可通过少量Krylov方法迭代高效求解。此外，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，且结合局部插值后可在整个PDE域内收敛。同时，原始约束方法仅适用于狄利克雷问题，而IBDL格式亦可处理诺伊曼边界条件。