  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.
