  Inspired by the numerical immersed boundary method, we introduce regularized
Stokes immersed boundary problems in two dimensions to describe regularized
motion of a 1-D closed elastic string in a 2-D Stokes flow, in which a
regularized $\delta$-function is used to mollify the flow field and singular
forcing. We establish global well-posedness of the regularized problems, and
prove that as the regularization parameter diminishes, string dynamics in the
regularized problems converge to that in the Stokes immersed boundary problem
with no regularization. Viewing the un-regularized problem as a benchmark, we
derive error estimates under various norms for the string dynamics. Our
rigorous analysis shows that the regularized problems achieve improved accuracy
if the regularized $\delta$-function is suitably chosen. This may imply
potential improvement in the numerical method, which is worth further
investigation.
