On convergence of the immersed boundary method for elliptic interface problems

Peskin’s Immersed Boundary (IB) method is 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∞ norm. But almost none rigorous proof could be found in the literature until recently [9] 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 the first order convergence with a log h factor of the IB method for elliptic interface problems essential without the boundary condition restrictions. The results should be applicable to the IB method for many different situations involving elliptic solvers for Stokes and Navier-Stokes equations. keywords: Immersed Boundary (IB) method, Dirac delta function, convergence of IB method, discrete Green function, discrete Green’s formula. AMS Subject Classification 2000 65M12, 65M20.