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\hypersetup{pdfauthor={Sida He},pdftitle={Stability and Convergence of a Regularized Immersed Boundary Method under Dynamic Cartesian Refinement. Part II: General Prescribed Motion},pdfsubject={Research manuscript v0.1.1: moving geometry, prescribed rigid-body motion, numerical verification and Lean-checked algebra},pdfkeywords={immersed boundary, adaptive mesh refinement, Stokes, convergence, Lean, manufactured solution}}
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\title{Stability and Convergence of a Regularized Immersed Boundary Method\protect\\under Dynamic Cartesian Refinement.\protect\\Part II: General Prescribed Motion}
\author{Sida He\\[3pt]
\small Huawei Technologies Co., Ltd.\\
\small\href{mailto:hesida1@huawei.com}{\texttt{hesida1@huawei.com}}\\
\small ORCID: \href{https://orcid.org/0000-0003-0849-6592}{0000-0003-0849-6592}}
\date{27 October 2026\\\small Research preprint, version 0.1.1}
\begin{document}
\maketitle
\noindent{\small\textbf{Disclosure:} Primarily AI-generated text. The author reports no understanding of the proofs; see the disclosure at the end of this manuscript.}\par\medskip
\begin{abstract}
We study a regularized immersed boundary discretization of unsteady Stokes flow around a rigid body with prescribed translation and rotation. The interface may change its position and orientation, while remaining uniformly separated from the outer wall. The fluid spaces are conforming, exactly divergence free, and constructed on fixed physical Cartesian grids with a retained coarse space and arbitrarily changing local enrichments. The method uses regularized surface-moment constraints, backward Euler time stepping, and current-space orthogonal projection. We construct a time-dependent coarse lifting that satisfies every discrete rigid-velocity constraint exactly. A moving collar approximation has a small spatial error in $H^1$ and a small Eulerian time-derivative error in $L^2$, although its time derivative need not be accurate in $H^1$. This distinction permits a quantitative convergence proof without a restriction coupling the time step to the number of mesh changes. A shifted-interface trace estimate accounts for the motion of the reaction surface during each time step and for physical slip between time nodes. Under stated piecewise regularity and an $L^2$ interface-reaction assumption, the energy error is bounded by the square roots of the retained coarse size, regularization width, surface-panel size, and maximum time step, together with a specified load-consistency error. The analysis assumes exact integration and does not prove adaptive optimality or cover fluid-driven motion. Manufactured moving-ellipse computations and a Lean algebraic companion test specified implementation and proof components.
\end{abstract}
\noindent\textbf{Keywords:} immersed boundary method; prescribed rigid motion; dynamic Cartesian refinement; moving interface; Stokes equations; regularization; convergence.

\input{sections/introduction}
\input{sections/setting}
\input{sections/trace}
\input{sections/approximation}
\input{sections/lifting}
\input{sections/energy}
\input{sections/convergence}
\input{numerics/numerical_section}
\FloatBarrier
\input{sections/discussion}
\input{sections/formal_status}
\input{sections/publication_disclosure}

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\end{document}