Convergence study of IB methods for Stokes equations with no-slip boundary conditions

TL;DR

This paper provides a rigorous convergence analysis of the Immersed Boundary method for Stokes equations with no-slip boundary conditions, establishing specific convergence rates for velocity and pressure in two dimensions.

Contribution

It offers the first rigorous proof of convergence rates for IB methods applied to Stokes equations with no-slip boundaries, introducing new analytical tools and a simple discrete delta function.

Findings

Pressure converges at O(√h|log h|) in L^2 norm.

Velocity converges at O(h|log h|) in infinity norm.

The analysis applies to various boundary conditions for well-posed problems.

Abstract

Peskin's Immersed Boundary (IB) model and method are among one of the most important 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 a convergence order $O(\sqrt{h} |\log h| )$ 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 and Neumann boundary conditions, a new discovered simplest $L^2$ discrete delta function, and the convergence proof of the IB method for elliptic interface problems \cite{li:mathcom}. The conclusion in this paper can apply to problems with different boundary conditions as long as the problems are wellposed. The proof process also provides an efficient way to decouple the system into three Helmholtz/Poisson equations without affecting the order of convergence. A non-trivial numerical example is also provided to confirm the theoretical analysis and the simple new discrete delta function.