Immersed Boundary Double Layer method: : An introduction of methodology on the Helmholtz equation

The Immersed Boundary (IB) method of Peskin (1977) [1] 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.