The Immersed Boundary Double Layer (IBDL) Method
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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 grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. The IB method has also been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces that 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 time step, this method can be prohibitively inefficient without preconditioning. This dissertation introduces a new, well-conditioned IB formulation for boundary value problems, called the Immersed Boundary Double Layer (IBDL) method. We formulate it for Poisson, Helmholtz, Brinkman, Stokes, and Navier-Stokes equations and demonstrate its efficiency over the other constraint method. In this new 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 the 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 boundary conditions.
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