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.

翻译：浸没边界（IB）方法已被广泛应用于模拟各类生物与工程应用中的流体-膜相互作用。尽管该方法的数值收敛性已通过经验验证，但由于控制方程中存在奇异强迫项，其理论证明尚未完成。本文的研究动机源于IB方法的一种特定变体，其中流体维度比浸没结构维度高两维。在这类余维数为2的问题中，浸没边界在连续公式中必然需要进行磨光处理。本文利用这一特性，证明了IB方法在应用于完全非线性流体中运动的弹性束缚粒子时的收敛性。