Convergence of the immersed-boundary finite-element method for the Stokes problem

TL;DR

This paper establishes convergence results for the immersed boundary finite element method applied to the Stokes problem, providing error estimates and validating them through numerical experiments.

Contribution

It offers the first rigorous convergence analysis of the immersed boundary finite element method for the Stokes problem, including error bounds in various norms.

Findings

Error estimate of order $h^{1-eta}$ in the $W^{1,1}$ norm for velocity and pressure.

Error estimate of order $h^{1-eta}$ in the $L^r$ norm for velocity.

Numerical examples confirm the theoretical error bounds.

Abstract

Convergence results for the immersed boundary method applied to a model Stokes problem with the homogeneous Dirichlet boundary condition are presented. As a discretization method, we deal with the finite element method. First, the immersed force field is approximated using a regularized delta function and its error in the $W^{-1,p}$ norm is examined for $1\le p<n/(n-1)$, $n$ being the space dimension. Then, we consider the immersed boundary discretization of the Stokes problem and study the regularization and discretization errors separately. Consequently, error estimate of order $h^{1-\alpha}$ in the $W^{1,1}\times L^1$ norm for the velocity and pressure is derived, where $\alpha$ is an arbitrarily small positive number. Error estimate of order $h^{1-\alpha}$ in the $L^r$ norm for the velocity is also derived with $r=n/(n-1-\alpha)$. The validity of those theoretical results are confirmed by numerical examples.