Regularized Stokes Immersed Boundary Problems in Two Dimensions: <scp>Well‐Posedness</scp>, Singular Limit, and Error Estimates
Jiajun Tong
Abstract
Abstract 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 δ ‐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 unregularized 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 δ ‐function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation. © 2020 Wiley Periodicals LLC