Convergence study of IB methods for Stokes equations with prescribed velocity boundary conditions

Abstract Peskin’s Immersed Boundary (IB) model and method are among the most important modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity for incompressible Stokes and Navier–Stokes equations. The convergence of the IB method for Stokes equations with periodic boundary conditions is discussed in Liu & Mori (2014, $L^p$ convergence of the immersed boundary method for stationary Stokes problems. SIAM J. Numer. Anal., 52, 496–514) and Mori (2008, Convergence proof of the velocity field for a Stokes flow immersed boundary method. Comm. Pure Appl. Math., 61, 1213–1263). However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has a convergence order $O(\sqrt{h})$ in the $L^{2}$ norm, while the velocity has an $O(h |\!\log h| )$ convergence order in the infinity norm in two-space dimensions. The proofs are based on splitting the singular source terms, discrete Green functions on finite lattices with homogeneous Dirichlet and Neumann boundary conditions, properties of discrete delta functions and the convergence proof of the IB method for elliptic interface problems (Li (2015) On convergence of the immersed boundary method for elliptic interface problems. Math. Comp., 84, 1169–1188). The conclusion in this paper can apply to problems with different boundary conditions as long as the problems are well-posed. The proof process also provides an efficient way to decouple the Stokes equations into three Helmholtz/Poisson equations without affecting the order of convergence. Nontrivial numerical examples are also provided to confirm the theoretical analysis.