L^p convergence of the immersed boundary method for stationary Stokes problems

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 < p < ∞) norm. One interesting consequence of our analysis is that the asymptotic error rates in the L¹ 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.