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

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.