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Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well‐Posedness , Singular Limit, and Error Estimates
Author(s) -
Tong Jiajun
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21968
Subject(s) - mathematics , regularization (linguistics) , stokes flow , mathematical analysis , string (physics) , boundary (topology) , immersed boundary method , navier–stokes equations , limit (mathematics) , boundary value problem , flow (mathematics) , physics , geometry , mathematical physics , compressibility , computer science , mechanics , artificial intelligence
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