On Convergence of the Immersed Boundary Method for Elliptic Interface Problems
NORTH CAROLINA STATE UNIV AT RALEIGH CENTER FOR RESEARCH IN SCIENTIFIC COMPUTATION
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Peskins 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. Peskins IB method is associated with discrete delta functions. It is believed that the IB method is first order accurate in the Linfinity 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.
- Numerical Mathematics