The Immersed Boundary (IB) method of Peskin (1977) is useful for problems that involve 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 therefore require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each step in time, this method can be prohibitively inefficient without preconditioning. This work introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. In order to lay the groundwork for similar formulations of Stokes and Navier-Stokes equations, this paper focuses on the Poisson and Helmholtz equations to introduce the methodology and 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 without preconditioning. Furthermore, the iteration count is independent of both the mesh size and spacing of the immersed boundary points. 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 boundary conditions.

中文翻译：

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浸入边界双层法：亥姆霍兹方程方法介绍

Peskin (1977) 的浸入边界 (IB) 方法对于涉及流体-结构相互作用或复杂几何形状的问题非常有用。通过使用独立于几何形状的规则笛卡尔网格，IB 框架产生了一个强大的数值方案，可以有效地处理浸入式可变形结构。此外，IB 方法已适用于具有给定边界数据的规定运动和其他偏微分方程的问题。解决这些问题的 IB 方法传统上涉及仅近似满足边界条件的罚力，或者将它们表述为约束问题。在后一种方法中，必须通过求解对应于不良条件的第一类积分方程的方程来找到未知力。因此，该操作可能需要 Krylov 方法的大量迭代，并且由于时间相关问题需要在每个步骤及时求解，因此在没有预处理的情况下，该方法的效率可能会非常低。这项工作为边值问题引入了一种新的、条件良好的 IB 公式，称为浸入式边界双层 (IBDL) 方法。为了为斯托克斯和纳维-斯托克斯方程的类似公式奠定基础，本文重点介绍泊松和亥姆霍兹方程，介绍该方法并证明其相对于原始约束方法的效率。在此双层公式中，未知边界分布的方程对应于条件良好的第二类积分方程，无需预处理即可通过 Krylov 方法的少量迭代有效求解。此外，迭代计数与网格大小和浸入边界点的间距无关。该方法远离边界收敛，当与局部插值结合时，它在整个 PDE 域内收敛。此外，虽然原始约束方法仅适用于狄利克雷问题，但 IBDL 公式也可用于诺依曼边界条件。