\section{Manufactured-solution verification}\label{sec:numerics}
We give a small numerical verification study of the specified conforming
method. It checks an implementation against an exact nonstationary
solution, following the manufactured-solution principle
\citep{salari2000}. The first five subsections concern a stationary
body; Section~\ref{sec:rotation-numerics} then verifies prescribed
rotation of the same fixed geometric interface.
The tests address bulk error, the physical interface trace, and mesh
transfer; they are not benchmarks for a conventional MAC implementation.

\subsection{An exact solution with nonzero interface shear}
Let $B$ be the disk of radius $R=0.2$ centered at $c=(1/2,1/2)$ in
$\Omega=(0,1)^2$. Write
\[
 X=x-c_1,\quad Y=y-c_2,\quad s=X^2+Y^2-R^2,\quad
 Q=x^2(1-x)^2y^2(1-y)^2,\quad s_+=\max(s,0).
\]
With $A=1000$, $\nu=0.1$, and $T=0.5$, define
\begin{equation}\label{eq:manufactured}
 \psi_0=AQ s_+^2,\qquad U=\curl\psi_0,\qquad
 u(x,t)=a(t)U(x),\qquad a(t)=1-e^{-t},\qquad p=0.
\end{equation}
The velocity is divergence free, vanishes inside $B$, and satisfies
no-slip on both $\Gamma$ and the outer square. Its initial value is zero.
It is smooth on either side of $\Gamma$ and globally $H^1$, so it has
the regularity required by Theorem~\ref{thm:rate}.

The ordinary bulk force is obtained by differentiating the exterior
polynomial, not from a discrete residual:
\begin{equation}\label{eq:manufactured-force}
 f(x,t)=e^{-t}U-\nu(1-e^{-t})\Delta U\quad(x\in F),
 \qquad f=0\quad(x\in B).
\end{equation}
For explicit evaluation, set
\[
 L=s^2\Delta Q+8s(XQ_x+YQ_y)+(16s+8R^2)Q.
\]
In $F$, $\Delta U=A(L_y,-L_x)$. For the normal
$n=(\cos\theta,\sin\theta)$ pointing out of the disk, the reaction in
\eqref{eq:traction-identity} is
\begin{equation}\label{eq:manufactured-traction}
 \lambda=\nu\partial_nu
 =8\nu A a(t)R^2Q(\sin\theta,-\cos\theta)\quad\hbox{on }\Gamma.
\end{equation}
Thus the interface shear is nonzero. The solution does not vanish in an
open collar of the interface, and its first spatial derivatives jump
across $\Gamma$. No surface delta term is added to the bulk load in
\eqref{eq:manufactured-force}; the constraint supplies the interface
reaction. Independent symbolic differentiation and integration by parts
check the forcing and the sign in \eqref{eq:manufactured-traction}.

The load is averaged exactly in time. On $[t_{n-1},t_n]$, with
$d_n=(e^{-t_{n-1}}-e^{-t_n})/\tau_n$, it is
$\bar f^n=d_nU-\nu(1-d_n)\Delta U$ in $F$.
The implementation evaluates the exponential difference using
\texttt{expm1}.

\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{numerics/figures/manufactured_geometry.pdf}
\caption{Left: one enrichment pattern for $M=16$; the second uses the
opposite side of the interface. Blue cells have the four additional
center degrees of freedom. The dashed circle bounds one kernel support.
The background discretization also extends through the white body.
Right: speed of the exact solution at $T=0.5$.}
\label{fig:numerical-geometry}
\end{figure}

\subsection{Discrete realization and numerical accuracy}
The implementation uses exactly the spaces of Section~\ref{sec:spaces}.
All coarse and local enrichment functions are represented in one fixed
hierarchical master basis. If $z_K$ is the center of a coarse cell, set
\[
 \mathcal R_h=\{K:\bigl||z_K-c|-R\bigr|<1.5h\},
\]
and alternate the two patches
\[
 \mathcal A_0=\{K\in\mathcal R_h:(z_K)_1<c_1+h/4\},\qquad
 \mathcal A_1=\{K\in\mathcal R_h:(z_K)_1>c_1-h/4\}.
\]
Both retain the complete coarse space. Their boundaries intersect the
regularization region; the two patterns are not nested.

For the regularizer, let
\[
 k(z)=\frac{\exp[-1/(1-z^2)]\,\mathbf1_{|z|<1}}
              {\int_{-1}^1\exp[-1/(1-r^2)]\,dr},\qquad
 \rho_\eps(x,y)=\frac{2}{\eps^2}
       k\left(\frac{\sqrt2x}{\eps}\right)
       k\left(\frac{\sqrt2y}{\eps}\right).
\]
This is a smooth even kernel supported in a square contained in the
radius-$\eps$ disk. Unless stated otherwise, $\eps=0.75h$ and the circle
has $P=\lceil2\pi R/h\rceil$ equal arclength panels, so
$\ell=2\pi R/P\le h$.

Bulk mass and viscosity matrices use four-point Gauss rules per
coordinate on the $h/2$ master cells. These rules integrate the
polynomial forms exactly, up to floating-point roundoff. The force and
error integrals respect the exact circular cut: cells are split into
exterior intervals using the circle equation, with additional breaks
at circle--cell-edge intersections. Uncut cells use order 10, and cut
intervals use order 24. The convolution integral is split at every
master-grid line before using order 16 in each coordinate; surface
moments use order 10 on each panel. Physical trace integration is
independent of the constraint residual and splits the circle into arcs
at all master-grid crossings.

The constraints are solved through an orthonormal basis for their row
space and a Schur complement. The singular-value cutoff is $10^{-12}$
relative to the largest singular value. No constraint is discarded in
any reported experiment: all matrices have full row rank, with a minimum
observed singular-value ratio of $6.47\times10^{-4}$. Changing the cutoff
to $10^{-10}$ or $10^{-14}$ at $M=16$ leaves the solution unchanged.
This observation is not a mesh-uniform multiplier stability claim.

The step right-hand side uses the cross-mesh mass pairing with the old
velocity. By orthogonality, this is equivalent to the constrained
$L^2$ projection in \eqref{eq:scheme}; it does not use nodal interpolation.
All energy diagnostics use the same physical mass matrix. The numerical
surface integrals are approximations to the exact moments assumed in
the theory. Consequently, the computations are a controlled approximate
realization, and the quadrature checks below do not replace a
quadrature-consistency theorem.

\subsection{Velocity convergence and physical slip}
Let $U_h$ be the piecewise constant reconstruction. We report
\begin{align*}
 E_0&=\frac{\norm{u_h^N-u(T)}_{L^2(\Omega)}}{\norm{u(T)}_{L^2(\Omega)}},
 &E_1&=\frac{\norm{U_h-u}_{L^2(0,T;U)}}{\norm u_{L^2(0,T;U)}},\\
 D_\Gamma&=\norm{\gamma_\Gamma U_h}_{L^2(0,T;L^2(\Gamma))}.
 \end{align*}
The volume norms include the body interior. In $E_1$, the exponential
time dependence of the exact solution is integrated analytically on
each interval; it is not replaced by its value at the right endpoint.

Table~\ref{tab:numerical-convergence} uses $N=400$ uniform time steps
and switches patch at every step. The three spatial lengths $h$, $\eps$,
and $\ell$ decrease together. Both velocity errors and physical slip
decrease. These four fixed-time-step results illustrate convergence;
they are not an asymptotic verification of the joint limit in
Theorem~\ref{thm:convergence}, nor do they establish a sharp spatial
rate. In particular, a rate faster than the half-order upper bound is
not a contradiction of Theorem~\ref{thm:rate}.

\begin{table}[tbp]
\centering
\caption{Velocity errors and physical no-slip defect, $N=400$. Errors
$E_0,E_1$ are expressed as percentages; $D_\Gamma$ is an absolute norm.}
\label{tab:numerical-convergence}
\input{numerics/results/convergence_rows.tex}
\end{table}

\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{numerics/figures/convergence.pdf}
\caption{The data in Table~\ref{tab:numerical-convergence}. Physical slip
is evaluated on the sharp circle, rather than inferred from the
regularized constraint residual.}
\label{fig:numerical-convergence}
\end{figure}

\subsection{Mesh switching, position sensitivity, and energy}
At $M=32$, keep $N=400$, $\eps$, and the panel partition fixed. Compare
a fixed patch with switching every 20, 5, or 1 steps. This changes only
the adaptation history. Table~\ref{tab:switching} shows that the final
relative velocity error stays between $2.31271\%$ and $2.31398\%$.
The accumulated projection losses are positive: the switches genuinely
discard resolved velocity components. The result supports robustness
for these histories; it does not assert identical errors for arbitrary
histories or replace the event-count-independent theorem.

\begin{table}[tbp]
\centering
\caption{Switching frequency at fixed $M=32$ and $N=400$. The last
column is $\sum_n\norm{(I-P^n)u^{n-1}}_0^2$.}
\label{tab:switching}
\input{numerics/results/frequency_rows.tex}
\end{table}

At $M=24$, shifting the fixed disk center to
$c=(1/2+\alpha h,1/2)$ with $\alpha\in\{0,1/4,1/2,3/4\}$ gives
Table~\ref{tab:position}. The manufactured force is recomputed for each
center. The errors remain of comparable size, with no anomalous growth
in these four alignments. Moving the disk also changes the exact
solution relative to the outer walls, so this is a position-sensitivity
test, not an isolated test of matrix conditioning.

\begin{table}[tbp]
\centering
\caption{Four fixed interface positions, $M=24$, $N=400$.}
\label{tab:position}
\input{numerics/results/position_rows.tex}
\end{table}

Across all 26 saved experiment runs, the largest absolute residual of
\eqref{eq:energy} is $1.87\times10^{-17}$, that of the jump
decomposition \eqref{eq:lossdecomposition} is $2.17\times10^{-19}$,
and the largest assembled panel-average constraint residual is
$5.33\times10^{-17}$. These algebraic checks would not by themselves
establish consistency with the physical boundary condition.

A separate unforced test uses $M=16$, $N=200$, initial data equal to
the constrained projection of $U$ in \eqref{eq:manufactured}, and a
patch change at every step. Its kinetic energy decreases at every
step, ending at $1.81\times10^{-7}$ times its initial value. This test
has different data from the manufactured problem; no exact-solution
error is assigned to it.

\begin{figure}[tbp]
\centering
\includegraphics[width=\linewidth]{numerics/figures/switching_energy.pdf}
\caption{Left: the fixed-time-step switching experiment. Right:
kinetic energy in the separate unforced decay test.}
\label{fig:switching-energy}
\end{figure}

\subsection{Checks against quadrature and temporal contamination}
For the $M=16$, $N=200$ dynamic case, increasing the convolution,
surface, and cut-cell orders from $(16,10,24)$ to $(24,16,32)$ changes
the final velocity by $5.67\times10^{-8}$ of the exact $L^2$ norm
and $2.17\times10^{-7}$ of the exact $H^1$ seminorm. These changes
are much smaller than the discretization errors. An independently
assembled integration-by-parts identity using
\eqref{eq:manufactured-traction} also checks the body-force assembly.

To isolate time discretization, keep the $M=32$ patch fixed and use
$N=100,200,400,800$. Relative $L^2$ differences between successive
final-time solutions are $1.9766\times10^{-3}$,
$9.8702\times10^{-4}$, and $4.9319\times10^{-4}$. Their ratios,
$2.0026$ and $2.0013$, are consistent with first-order time stepping.
These are temporal self-convergence differences, not errors against
the exact semidiscrete solution. In the finest dynamic spatial case,
doubling $N$ from 400 to 800 changes $E_0$ from $1.13535\%$ to
$1.12074\%$; thus the reported error is not predominantly temporal.
That last comparison also changes the patch-switching frequency and
is not used to infer a temporal order.

The companion provides the solver, exact-solution verifier, independent
implementation checks, all 26 parameter files, final coefficient vectors,
and diagnostic histories,
and scripts that regenerate the tables and figures. The results support
the implementation and the stated qualitative trends. They do not prove
adaptive optimality, multiplier convergence, or general robustness of
other IB discretizations.\FloatBarrier