The immersed boundary (IB) method has been used as a means to simulate fluid-membrane interactions in a wide variety of biological and engineering applications. Although the numerical convergence of the method has been empirically verified, it is theoretically unproved because of the singular forcing terms present in the governing equations. This paper is motivated by a specific variant of the IB method, in which the fluid is 2 dimensions greater than the dimension of the immersed structure. In these co-dimension 2 problems the immersed boundary is necessarily mollified in the continuous formulation. In this paper we leverage this fact to prove convergence of the IB method as applied to a moving elastically bound particle in a fully non-linear fluid.

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