On convergence of the immersed boundary method for elliptic interface problems

Li, Zhilin

doi:10.1090/S0025-5718-2014-02932-3

On convergence of the immersed boundary method for elliptic interface problems
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Math. Comp. 84 (2015), 1169-1188 Request permission

Abstract:

Peskin’s Immersed Boundary (IB) method has been one of the most popular numerical methods for many years and has been applied to problems in mathematical biology, fluid mechanics, material sciences, and many other areas. Peskin’s IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the $L^{\infty }$ norm. But almost no rigorous proof could be found in the literature until recently [Mori, Comm. Pure. Appl. Math: 61:2008] in which the author showed that the velocity is indeed first order accurate for the Stokes equations with a periodic boundary condition. In this paper, we show first order convergence with a $\log h$ factor of the IB method for elliptic interface problems with Dirichlet boundary conditions.

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