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

Abstract

Abstract.In this paper, we analyze the convergence of the immersed boundary (IB) method as applied to a static Stokes flow problem. Using estimates obtained in [5], we consider a problem in which a d-dimensional structure is immersed in n-dimension, 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 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.Key words. immersed boundary method, L p error estimates, discrete delta functions.1. Introduction. The immersed boundary method has been widely used to solve problems with moving interfaces (fluid-structure interaction, two phase fluid flow, etc) along which variables of interest often possess discontinuities. In IB formulations, these problems are recast as partial differential equations (PDE) over a simpler domain with singular source terms distributed along the interfaces. Dirac delta functions are used to represent the singular source term as a distribution defined on the entire fluid domain. The PDE are often discretized on a Eulerian grid over the fluid domain, while the singular source term is often discretized by integrating over a Lagrangian grid on the interface using regularized Dirac delta functions (often called discrete delta functions) . The discretized equations are then often solved with standard methods.In addition to the IB method, many other methods are often used to solve such free surface problems, with boundary integral methods and level set methods being prime examples. Among these methods, algorithms based on the IB method are often very efficient and easy to implement. Although it has been justified by numerical results in practice, convergence of the IB method is often unresolved from an analytical point of view. Despite being the topic of many papers [6,8,9,1,3,4], convergence analysis of IB methods is still at a primary stage. As a first step to analyze full dynamic problems, convergence properties of the IB method were studied for a stationary Stokes flow problem in [6]. For the velocity field, point-wise and L ∞ error estimates are obtained. Then as an application of the results, L 2 error estimates are studied for a simple dynamic problem. The analysis relies on the geometric structure of the 2D model problem, which is a one-dimensional elastic string immersed in a two-dimensional fluid domain.In [5], the analysis is extended to a more general elliptic model problem. It is worth mentioning that the analysis in [5], despite carried through for an immersed boundary method setup, is developed for more general problems involving discontinuities that are regularized by using discrete delta functions. For an immersed boundary model problem with general choices of n and d, where n is the dimension of the fluid domain and d is the dimension of the immersed structure, the error is decomposed and carefully ...

Analysis of the immersed boundary method for a finite element Stokes problem

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