\section{Prescribed geometry and the Eulerian problem}\label{sec:setting}
Let $\Omega=(0,1)^2$ and let $B_*\subset\R^2$ be a bounded simply connected reference body with connected $C^6$ boundary. Write
\begin{equation}\label{eq:rigid}
 B(t)=c(t)+Q(\theta(t))B_*,\quad \Gamma(t)=\partial B(t),\quad
 F(t)=\Omega\setminus\overline{B(t)},\quad
 b(t,x)=\dot c(t)+\omega(t)J(x-c(t)),
\end{equation}
where $Q(\theta)$ is the planar rotation matrix, $J(x_1,x_2)=(-x_2,x_1)$, and $\omega=\dot\theta$. Assume $c,\theta\in C^2([0,T])$ and
$\dist(\overline{B(t)},\partial\Omega)\ge d_*>0$ for every $t$. All motion and geometry constants below may depend on these fixed data. They do not depend on a mesh history. This is one prescribed rigid body; there is no equation determining $c$ or $\theta$ from fluid forces.

All velocities are expressed at physical Eulerian points $x$. Set
\begin{equation}\label{eq:U}
 U=\{v\in H^1_0(\Omega)^2:\divg v=0\},\quad
 \norm v_U=\norm{\nabla v}_{L^2(\Omega)},\quad
 a(v,z)=\nu\ip{\nabla v}{\nabla z},\quad \nu>0.
\end{equation}
The bulk inner product and norm are denoted by $(\cdot,\cdot)$ and $\norm\cdot_0$. Define $V(t)=\{v\in U:\gamma_t v=0\}$, where $\gamma_t$ is the trace on $\Gamma(t)$. Unlike $U$, this constrained space changes with time.

We consider a sufficiently regular target velocity satisfying
\begin{equation}\label{eq:continuous}
 u(t)\in U,\qquad \gamma_t u(t)=b(t)|_{\Gamma(t)},\qquad
 (\partial_tu(t),v)+a(u(t),v)=\dual{f(t)}v\quad(v\in V(t)).
\end{equation}
For the quantitative theorem we assume the stronger reaction identity
\begin{equation}\label{eq:traction}
 (\partial_tu(t),v)+a(u(t),v)
 +\int_{\Gamma(t)}\lambda(t)\cdot\gamma_t v\,ds
 =\dual{f(t)}v\qquad(v\in U),
\end{equation}
with $\int_0^T\norm{\lambda(t)}_{L^2(\Gamma(t))}^2dt<\infty$.
The sign of $\lambda$ is defined by this identity. This is an assumption on the continuous reaction, not a uniform stability assumption on discrete multiplier rows. The pressure has been eliminated from the test space. No pressure or multiplier error estimate is asserted.

Equation~\eqref{eq:continuous} is the weak form of Eulerian unsteady Stokes flow, not a material-derivative equation. The transport calculus for evolving Hilbert spaces and the additional incompressibility issues in moving-domain fluid spaces are developed in \citet{alphonse2015,djurdjevac2023}. We do not invoke those existence results to infer the piecewise smoothness required here. The convergence theorem is conditional on a target solution with the explicitly stated regularity. Its comparison and lifting constructions are proved below.

The background-domain formulation must specify the solid interior. To represent the prescribed rigid velocity there, take $u_0=b(0)$ in $B(0)$ and, with interior pressure chosen to be zero, take
\begin{equation}\label{eq:interior-force}
 f_B(t,x)=\partial_t b(t,x)
 =\ddot c(t)+\dot\omega(t)J(x-c(t))-\omega(t)J\dot c(t).
\end{equation}
Since $\Delta b=0$ and $\divg b=0$, this gives the rigid interior solution. The difference from any other sufficiently regular interior solution has zero initial and boundary data. Its energy identity on $B(t)$ has no boundary transport contribution, since that difference vanishes on $\Gamma(t)$, and implies uniqueness. Thus the exterior restriction is the intended fluid solution. Formula~\eqref{eq:interior-force} is not the material rigid acceleration: a centrifugal term must not be inserted into this Eulerian Stokes model. The prescribed interior load is a modeling device, not an external force on a freely moving solid.

Choose a nonnegative, even $\rho\in C_c^\infty(\R^2)$, supported in the unit ball, with $\int\rho=1$, and set $\rho_\eps(x)=\eps^{-2}\rho(x/\eps)$. For the zero extension $Ev$ of $v\in U$, define
\begin{equation}\label{eq:J}
 J_{\eps,t}v=\gamma_t(\rho_\eps*Ev),\qquad
 b_{\eps,t}(v,\mu)=\int_{\Gamma(t)}\mu\cdot J_{\eps,t}v\,ds.
\end{equation}
This exact-convolution operator retains the interpolation/spreading adjoint pairing of regularized IB methods \citep{peskin2002}. Its integrals are not replaced by marker sampling in the analysis. Let $\mathcal P$ be an arclength partition of $\Gamma(t)$, let $\ell$ be its largest panel length, and let $\Pi_{\mathcal P}$ be the $L^2$ projector onto vector-valued panel constants. The inhomogeneous discrete constraint is
\begin{equation}\label{eq:constraint}
 \Pi_{\mathcal P}(J_{\eps,t}u_h-b(t))=0.
\end{equation}
Regularization width, panel partition and Cartesian enrichment may change independently at each time node. A bar denotes the maximum width used in the entire computation, including initialization.