Convergence study of IB methods for Stokes equations with non-periodic boundary conditions

Peskin's Immersed Boundary (IB) model and method are among the most popular modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity. 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 convergence order $O(\sqrt{h})$ in the $L^2$ norm while the velocity has $O(h)$ convergence in the infinity norm in two-dimensions (2D). The proofs are based on the idea of the immersed interface method, and the convergence proof of the IB method for elliptic interface problems \cite{li:mathcom}. The proof is intuitive and the conclusion can apply to different boundary conditions as long as the problem is well-posed. The proof process also provides an efficient way to decouple the system into three Helmholtz/Poisson equations without affecting the accuracy. A non-trivial numerical example is also provided to confirm the theoretical analysis.