Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well-posedness, Singular Limit, and Error Estimates

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 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 δ-function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation.

Stability of the Stokes Immersed Boundary Problem with Bending and Stretching Energy

“…They prove nonlinear stability of equilibrium states with explicit exponential decay estimates, and verify the optimality of which numerically. In a recent paper [37], Tong studies the regularized problem of this model, and derives error estimates under various norms. When the elastic membrane is surrounded by inviscid fluids, the model become to the one called the hydroelastic wave.…”