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The Immersed Boundary (IB) method is widely used for solving PDE on complex geometries or exploring problems that involve fluid-structure interactions. 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 the boundary conditions of immersed deformable structures. However, one approach for using the IB framework to solve problems involving rigid bodies corresponds to the numerical solution of a first-kind integral equation, which can have large condition numbers, weakening the efficiency of the method. Moreover, the method is only able to achieve first order accuracy. The Immersed Boundary Smooth Extension (IBSE) method achieves greater accuracy, but it relies on solving an extension problem with a poorly conditioned operator, requiring an expensive precomputation. In this talk, I'll discuss the Immersed Boundary Method and then give our new Double Layer formulation of the Immersed Boundary method, which corresponds to a better conditioned second-kind integral equation.

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