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On convergence of the immersed boundary method for elliptic interface problems

Author: Zhilin Li
Journal: Math. Comp. 84 (2015), 1169-1188
MSC (2010): Primary 65N06, 65M12, 65M15
Published electronically: December 16, 2014
MathSciNet review: 3315504
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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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Additional Information

Zhilin Li
Affiliation: Center for Research in Scientific Computation (CRSC) and Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695 — and — guest professor of School of Mathematical Sciences, Nanjing Normal University

Keywords: Immersed Boundary (IB) method, Dirac delta function, convergence of IB method, discrete Green function, discrete Green's formula
Received by editor(s): January 26, 2012
Received by editor(s) in revised form: March 1, 2013
Published electronically: December 16, 2014
Additional Notes: The author was supported in part by the AFSOR grant FA9550-09-1-0520, and the NIH grant 096195-01.
Article copyright: © Copyright 2014 American Mathematical Society