$L^p$ Convergence of the Immersed Boundary Method for Stationary Stokes Problems

Abstract

In this paper, we analyze the convergence of the immersed boundary method as applied to a static Stokes flow problem. Using estimates obtained in [Y. Liu and Y. Mori, SIAM J. Numer. Anal., 50 (2012), pp. 2986--3015], we consider a problem in which a $d$-dimensional structure is immersed in an $n$-dimensional domain, and prove error estimates for both the pressure and the velocity field in the $L^p(1\leq p\leq \infty)$ norm. One interesting consequence of our analysis is that the asymptotic error rates in the $L^1$ norm do not depend on either $d$ or $n$ and in the $L^p$ ($p>1$) norm they only depend on $n-d$. The resulting estimates are checked numerically for optimality.