\section{Prescribed rotation with fixed immersed geometry}
\label{sec:fixed-geometry}

The position of an immersed interface need not change when the material
on it moves. A disk rotating about its fixed center is a useful example:
its Eulerian boundary remains fixed, but its no-slip velocity is
nonzero and may vary in time. We prove an extension for this problem.
The common homogeneous comparison family from Section~\ref{sec:spaces}
is retained, and a second, explicitly constructed coarse-grid function
provides an exact lifting of every regularized boundary constraint.
Prescribed rigid motion in fictitious-domain formulations predates this
construction \citep{glowinski1997,kallemov2016}. The point here is the
simultaneous feasibility of one coarse lift for every admitted refinement,
regularization width and surface partition.

\subsection{An exact common lifting}
Let $B=\{x:|x-c|<R\}\Subset\Omega$ be a disk. Write
\[
 J=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad
 r(x)=J(x-c),\qquad \omega\in H^1(0,T).
\]
Thus the prescribed boundary velocity is $\omega(t)r$; positive
$\omega$ means counterclockwise rotation. Choose $d_*>0$ and
$\chi\in C_c^5(\Omega)$ such that $\chi=1$ on the open
$d_*$-neighborhood of $\overline B$. Set
\begin{equation}\label{eq:rotation-lift}
 \psi(x)=-\tfrac12\chi(x)|x-c|^2,
 \qquad g=\curl\psi,\qquad g_h=\curl I_h\psi.
\end{equation}
The sign is consistent with
$\curl\psi=(\partial_y\psi,-\partial_x\psi)$:
$g=r$ wherever $\chi=1$. The cutoff and $d_*$ are fixed independently
of the mesh and of the regularization parameters. The stated $C^5$
regularity suffices for $\psi\in H^4$ and $g\in W^{3,\infty}$;
a $C^\infty$ cutoff is also admissible. This includes the piecewise
polynomial $C^5$ cutoff used in the numerical verification.

\begin{lemma}[Common exact rotation lift]\label{lem:rotation-lift}
Suppose
\begin{equation}\label{eq:rotation-collar-condition}
 \bar\eps+\sqrt2h<d_*.
\end{equation}
Then $g_h\in\curl S_h\subset X_{h,\mathcal A}$ for every admissible
refinement patch, and
\begin{align}
 g_h&=r\quad\hbox{on }B\hbox{ and on every regularization support
 centered on }\Gamma,\nonumber\\
 \gamma_\Gamma g_h&=r,
 \qquad J_\eps g_h=r\quad(0<\eps\le\bar\eps),
 \label{eq:rotation-exact-trace}\\
 \norm{g_h-g}_{H^1(\Omega)}&\le C_g h^2.
 \label{eq:rotation-lift-error}
\end{align}
Consequently the nonempty affine constraint space is exactly
\begin{equation}\label{eq:rotation-affine-space}
 \begin{split}
 \omega_n g_h+Z^n
 =\{v_h\in X_{h,\mathcal A^n}:\ &
 \Pi_{\mathcal P^n}(J_{\eps_n}v_h-\omega_n r)=0\},\qquad
 \omega_n=\omega(t_n).
 \end{split}
\end{equation}
These assertions require neither independence nor an inf--sup bound
for the surface constraints.
\end{lemma}
\begin{proof}
Every coarse cell meeting the $\bar\eps$-neighborhood of $\Gamma$
lies inside the $(\bar\eps+\sqrt2h)$-neighborhood of $\overline B$.
The same is true of every coarse cell meeting $B$. On these cells,
$\psi=-|x-c|^2/2$, and bicubic Hermite interpolation reproduces this
quadratic polynomial exactly. This proves the first assertion and
the physical trace equality. All relevant kernel supports are inside
$\Omega$, so zero extension does not change the convolution there.
Normalization and evenness of $\rho$ imply
$\int\rho_\eps(z)\,dz=1$ and $\int z\rho_\eps(z)\,dz=0$. Hence, for
$x\in\Gamma$,
\[
 J_\eps g_h(x)
   =\int\rho_\eps(z)r(x-z)\,dz
   =r(x)-J\int z\rho_\eps(z)\,dz=r(x).
\]
This is affine reproduction by zeroth and first kernel moments, the
continuum counterpart of the familiar IB moment conditions
\citep{peskin2002,liumori2012}; the displayed equality proves the exact
property needed for our lift without a discrete-delta convergence theorem.
The interpolation estimate \eqref{eq:BFS-interpolation}, applied to
the fixed smooth $\psi$, gives \eqref{eq:rotation-lift-error}.
Finally subtracting $\omega_n g_h$ from either side of
\eqref{eq:rotation-affine-space} gives exactly the homogeneous
constraint defining $Z^n$.
\end{proof}

\subsection{Continuous and discrete centered equations}
Subtracting a lift is the standard reduction of nonhomogeneous essential
boundary data to a homogeneous variational problem; see, for example,
\citet[Chapter 33]{ern2021ii} for its elliptic form.
The time-dependent load and the common discrete lift needed here are
specified explicitly below, rather than inferred from that elliptic result.
Let $f\in L^2(0,T;U')$, and prescribe initial data in the form
$u_0=\omega(0)g+w_0$, where $w_0\in H$. Define
\begin{equation}\label{eq:rotation-continuous-load}
 \dual{\mathcal F(t)}v
 =\dual{f(t)}v-\omega'(t)\ip gv-\omega(t)a(g,v),
 \qquad v\in U.
\end{equation}
This functional belongs to $L^2(0,T;U')$. Let $w$ be the solution of
\eqref{eq:weak} with initial value $w_0$ and load $\mathcal F$, and set
\begin{equation}\label{eq:rotation-continuous-solution}
 G(t)=\omega(t)g,\qquad u(t)=w(t)+G(t).
\end{equation}
Then $u\in G+L^2(0,T;V)$, $u\in C([0,T];L^2)$,
$\gamma_\Gamma u=\omega r$ almost everywhere in time, and
\[
 \dual{\partial_tu}v+a(u,v)=\dual f v\qquad(v\in V).
\]
Conversely any solution in this affine class has the stated form by
subtracting $G$. Existence and uniqueness therefore follow from the
homogeneous problem, without solving a time-dependent lifting problem.

Initialize by the true affine projection of the physical initial data:
\begin{equation}\label{eq:rotation-initialization}
 \begin{split}
 u_h^0&=\omega_0g_h+P^0(u_0-\omega_0g_h),\\
 w_h^0&=P^0\bigl(w_0+\omega_0(g-g_h)\bigr).
 \end{split}
\end{equation}
The difference from the centered initialization $P^0w_0$ is bounded
by $C_gh^2|\omega_0|$ in the bulk norm. One may instead use that
centered initialization; all results below remain valid and the
initial perturbation in their proofs is then zero. Put
\[
 \delta\omega_n=\frac{\omega_n-\omega_{n-1}}{\tau_n}
        =\frac1{\tau_n}\int_{t_{n-1}}^{t_n}\omega'(t)\,dt,
 \qquad
 \bar\omega_n=\frac1{\tau_n}\int_{t_{n-1}}^{t_n}\omega(t)\,dt.
\]
The centered backward Euler scheme is
\begin{align}
 w_h^n&\in Z^n,\qquad u_h^n=w_h^n+\omega_ng_h,\nonumber\\
 \ip{(w_h^n-P^nw_h^{n-1})/\tau_n}{v_h}+a(w_h^n,v_h)
  &=\dual{\mathcal F_h^n}{v_h}\quad(v_h\in Z^n),
 \label{eq:rotation-scheme}\\
 \dual{\mathcal F_h^n}v
  &=\dual{f^n}v-\delta\omega_n\ip{g_h}v-\omega_n a(g_h,v).
 \label{eq:rotation-discrete-load}
\end{align}
The velocity exists uniquely by positive definiteness on $Z^n$.
It satisfies the exact nonhomogeneous moment constraints
\eqref{eq:rotation-affine-space} at every time level. Although $g_h$
belongs to every unconstrained Cartesian space, it generally does
\emph{not} belong to $Z^n$; no identity $P^ng_h=g_h$ is used.

\begin{remark}[Affine projection and a potentially inconsistent reset]
\label{rem:rotation-transfer}
For a fixed step the genuine $L^2$ projection onto the current affine
space is
\[
 \mathcal P_n^{\rm aff}q
    =\omega_ng_h+P^n(q-\omega_ng_h).
\]
Orthogonality gives
$\ip{\mathcal P_n^{\rm aff}q-q}{v_h}=0$ for every $v_h\in Z^n$.
Consequently \eqref{eq:rotation-scheme} is equivalent to the physical
velocity equation
\[
 \ip{(u_h^n-\mathcal P_n^{\rm aff}u_h^{n-1})/\tau_n}{v_h}
       +a(u_h^n,v_h)=\dual{f^n}{v_h},\qquad v_h\in Z^n.
\]
Indeed, writing $u_h^n=w_h^n+\omega_ng_h$ in this identity produces
$\delta\omega_n\ip{g_h}{v_h}$ in addition to the centered difference,
which moves to the right-hand side of
\eqref{eq:rotation-discrete-load}. A true affine projection therefore
retains the prescribed acceleration.

It is different from the lift-reset map
\[
 \mathcal R_nq=\omega_ng_h+P^n(q-\omega_{n-1}g_h),
 \qquad
 \mathcal P_n^{\rm aff}q
       =\mathcal R_nq-\tau_n\delta\omega_nP^ng_h.
\]
Replacing the affine projection by $\mathcal R_n$ in the displayed
physical equation, without a compensating acceleration term, omits
$-\delta\omega_n\ip{g_h}{v_h}$ from the centered load. It therefore
defines a different evolution when that functional is nonzero.
The centered formulation specifies the transfer and the lifting
derivative unambiguously.
\end{remark}

\subsection{Centered energy and the physical boundary defect}
Let
\[
 \mathcal D_\omega
 =\norm{w_0}_0+\norm f_{L^2(0,T;U')}
       +\norm{\omega'}_{L^2(0,T)}+\norm\omega_{L^\infty(0,T)}.
\]
Constants below may depend on the fixed lifting, $\nu$, and $T$, but
not on the patch history, panel partitions, regularization widths, or
the number of time steps.

\begin{proposition}[Centered energy and unchanged interface order]
\label{prop:rotation-energy}
At every step of \eqref{eq:rotation-scheme},
\begin{align}
 &\norm{w_h^n}_0^2-\norm{w_h^{n-1}}_0^2
   +\norm{w_h^n-w_h^{n-1}}_0^2+2\tau_na(w_h^n,w_h^n)
 \nonumber\\
 &\qquad=2\tau_n\bigl\{\dual{f^n}{w_h^n}
          -\delta\omega_n\ip{g_h}{w_h^n}
          -\omega_n a(g_h,w_h^n)\bigr\}.
 \label{eq:rotation-energy}
\end{align}
The increment has the orthogonal decomposition
\begin{equation}\label{eq:rotation-loss}
 \norm{w_h^n-w_h^{n-1}}_0^2
 =\norm{w_h^n-P^nw_h^{n-1}}_0^2
      +\norm{(I-P^n)w_h^{n-1}}_0^2.
\end{equation}
Moreover,
\begin{equation}\label{eq:rotation-energy-bound}
 \max_n\norm{w_h^n}_0^2
   +\sum_n\tau_n\norm{w_h^n}_U^2
   +\sum_n\norm{w_h^n-w_h^{n-1}}_0^2
 \le C\mathcal D_\omega^2.
\end{equation}
Let $U_h(t)=u_h^n$, $W_h(t)=w_h^n$, and $\omega_\tau(t)=\omega_n$
on $(t_{n-1},t_n]$. Then
\begin{align}
 \norm{\gamma_\Gamma U_h-\omega_\tau r}_{L^2(0,T;L^2(\Gamma))}
 &\le C(\sqrt{\bar\eps}+\sqrt{\bar\ell})\mathcal D_\omega,
 \label{eq:rotation-slip-nodal}\\
 \norm{\gamma_\Gamma U_h-\omega r}_{L^2(0,T;L^2(\Gamma))}
 &\le C(\sqrt{\bar\eps}+\sqrt{\bar\ell})\mathcal D_\omega
   +\tau_{\max}\norm r_{L^2(\Gamma)}\norm{\omega'}_{L^2(0,T)}.
 \label{eq:rotation-slip-continuous}
\end{align}
In particular there is no spatial lifting-error term in these trace
bounds.
\end{proposition}
\begin{proof}
Test \eqref{eq:rotation-scheme} with $w_h^n$. Projection orthogonality
and the polarization identity give \eqref{eq:rotation-energy}, and
the two components in \eqref{eq:rotation-loss} are orthogonal.
The lift estimate gives a uniform bound on $\norm{g_h}_U$ and
$\norm{g_h}_0$. Jensen's inequality and Poincare's inequality imply
\[
 \left(\sum_n\tau_n\norm{\mathcal F_h^n}_{U'}^2\right)^{1/2}
 \le \norm f_{L^2(0,T;U')}
       +C_P\norm{g_h}_0\norm{\omega'}_{L^2(0,T)}
       +\nu\sqrt T\norm{g_h}_U\norm\omega_{L^\infty(0,T)}.
\]
The initial norm is at most
$\norm{w_0}_0+C_gh^2|\omega_0|$. Summation of the energy identity and
Young's inequality now prove \eqref{eq:rotation-energy-bound}. Since
$\gamma_\Gamma U_h-\omega_\tau r=\gamma_\Gamma W_h$ exactly,
Lemma~\ref{lem:trace} and the centered energy bound prove
\eqref{eq:rotation-slip-nodal}. Finally the fundamental theorem of
calculus and Cauchy--Schwarz give
\begin{equation}\label{eq:omega-time-bounds}
 \norm{\omega_\tau-\omega}_{L^2(0,T)}
       \le\tau_{\max}\norm{\omega'}_{L^2(0,T)},\qquad
 \norm{\omega_\tau-\omega}_{L^\infty(0,T)}
       \le\sqrt{\tau_{\max}}\norm{\omega'}_{L^2(0,T)}.
\end{equation}
The triangle inequality proves \eqref{eq:rotation-slip-continuous}.
\end{proof}

\begin{proposition}[Background kinetic energy and constraint work]
\label{prop:rotation-background-energy}
Set $G_h^n=\omega_ng_h$ and define the work functional at the computed
velocity by
\begin{equation}\label{eq:rotation-constraint-work}
 \mathcal W_\Gamma^n
 =\ip{(u_h^n-u_h^{n-1})/\tau_n}{G_h^n}
       +a(u_h^n,G_h^n)-\dual{f^n}{G_h^n}.
\end{equation}
Then the physical-velocity variable satisfies the exact
background-domain identity
\begin{align}
 &\norm{u_h^n}_0^2-\norm{u_h^{n-1}}_0^2
   +\norm{u_h^n-u_h^{n-1}}_0^2+2\tau_na(u_h^n,u_h^n)
 \nonumber\\
 &\qquad=2\tau_n\dual{f^n}{u_h^n}
                   +2\tau_n\mathcal W_\Gamma^n.
 \label{eq:rotation-background-energy}
\end{align}
The value of $\mathcal W_\Gamma^n$ is unchanged if $G_h^n$ is
replaced by any other lift in the same current affine constraint
space. If a multiplier formulation is used with sign convention
\begin{equation}\label{eq:rotation-mixed-residual}
 \ip{(u_h^n-u_h^{n-1})/\tau_n}{v_h}+a(u_h^n,v_h)
       +b_{\eps_n}(v_h,\lambda_h^n)=\dual{f^n}{v_h}
       \quad(v_h\in X_{h,\mathcal A^n}),
\end{equation}
then $\mathcal W_\Gamma^n=-b_{\eps_n}(G_h^n,\lambda_h^n)$.
This scalar is unique even if the multiplier is not.
\end{proposition}
\begin{proof}
Let
\[
 R_n(v)=\ip{(u_h^n-u_h^{n-1})/\tau_n}v
                  +a(u_h^n,v)-\dual{f^n}v.
\]
The centered scheme and projection orthogonality give $R_n(v_h)=0$
for $v_h\in Z^n$. Since $u_h^n-G_h^n=w_h^n\in Z^n$,
$R_n(u_h^n)=R_n(G_h^n)=\mathcal W_\Gamma^n$. Multiplication by
$2\tau_n$ and the polarization identity prove
\eqref{eq:rotation-background-energy}. Any two lifts in the current
affine space differ by an element of $Z^n$, on which $R_n$ vanishes,
proving lifting independence. In finite dimensions, $R_n|_{Z^n}=0$
also implies that $-R_n$ is in the range of the transpose of the
constraint map. Hence a multiplier satisfying
\eqref{eq:rotation-mixed-residual} exists, without a row-rank
assumption. Testing that equation with $G_h^n$ gives the stated
formula. The difference of two such multipliers pairs to zero with
every vector in $X_{h,\mathcal A^n}$, including $G_h^n$, so it cannot
change $\mathcal W_\Gamma^n$.
\end{proof}

\subsection{Reduction to the homogeneous convergence theorem}
The discrete load in \eqref{eq:rotation-discrete-load} is not exactly
the time average of \eqref{eq:rotation-continuous-load}. The next
estimate controls this difference independently of the mesh history.

\begin{lemma}[Lifting load perturbation]\label{lem:rotation-load}
Let $\overline{\mathcal F}^{\,n}$ denote the exact time average of
$\mathcal F$ on the $n$th interval, and put
$\mathcal E_h^n=\mathcal F_h^n-\overline{\mathcal F}^{\,n}$. Then
\begin{align}
 \dual{\mathcal E_h^n}v
  ={}&-\delta\omega_n\ip{g_h-g}v
       -\omega_n a(g_h-g,v)
       -(\omega_n-\bar\omega_n)a(g,v),
 \label{eq:rotation-load-defect}\\
 \left(\sum_n\tau_n\norm{\mathcal E_h^n}_{U'}^2\right)^{1/2}
 &\le C\left[
 h^2\{\norm{\omega'}_{L^2(0,T)}+\norm\omega_{L^\infty(0,T)}\}
       +\tau_{\max}\norm{\omega'}_{L^2(0,T)}\right].
 \label{eq:rotation-load-bound}
\end{align}
\end{lemma}
\begin{proof}
Time averaging \eqref{eq:rotation-continuous-load} and subtracting
gives \eqref{eq:rotation-load-defect}. Its first two terms are bounded
in $U'$ by
$C_Ph^2C_g|\delta\omega_n|$ and
$\nu C_gh^2|\omega_n|$. For the last term use
\[
 \sum_n\tau_n|\omega_n-\bar\omega_n|^2
       \le\tau_{\max}^2\norm{\omega'}_{L^2(0,T)}^2,
\]
which follows interval by interval from the fundamental theorem of
calculus, as in \eqref{eq:time-residual}. Jensen's inequality for
$\delta\omega_n$, the bound
$\sum_n\tau_n|\omega_n|^2\le T\norm\omega_{L^\infty}^2$, and the
triangle inequality in the weighted sequence norm prove the result.
\end{proof}

\begin{theorem}[Strong convergence for prescribed fixed-disk rotation]
\label{thm:rotation-convergence}
Let $f\in L^2(0,T;U')$, $\omega\in H^1(0,T)$, and $w_0\in H$.
Use the common lift and initialization \eqref{eq:rotation-initialization}.
Suppose
\eqref{eq:parameters-to-zero} holds and, for all sufficiently large
$m$, \eqref{eq:rotation-collar-condition} holds. The admissible patch,
kernel width, and surface panel partition may change at every step.
Then the physical velocities satisfy
\begin{equation}\label{eq:rotation-convergence}
 \norm{U_m-u}_{L^2(0,T;U)}\longrightarrow0,
 \qquad \sup_{0\le t\le T}\norm{U_m(t)-u(t)}_0\longrightarrow0.
\end{equation}
Also the sum of the two centered losses in
\eqref{eq:rotation-loss} tends to zero. None of these statements
requires a regular interface multiplier or a restriction on the number
of mesh changes.
\end{theorem}
\begin{proof}
For each run let $y_h^n\in Z^n$ solve the homogeneous scheme
\eqref{eq:scheme} with initial value $P^0w_0$ and exact averaged load
$\overline{\mathcal F}^{\,n}$. Theorem~\ref{thm:convergence} applies
to this auxiliary computation and the fixed continuous data
$(w_0,\mathcal F)$. Its reconstruction $Y_m$ converges strongly to
$w$ in both norms stated in \eqref{eq:strong-convergence}, and its
total loss tends to zero.

Set $e_h^n=w_h^n-y_h^n$. Subtraction gives
\[
 \ip{(e_h^n-P^ne_h^{n-1})/\tau_n}{v_h}+a(e_h^n,v_h)
       =\dual{\mathcal E_h^n}{v_h},\qquad
 e_h^0=\omega_0P^0(g-g_h).
\]
The same energy argument as in Proposition~\ref{prop:energy} yields
\begin{equation}\label{eq:rotation-perturbation-stability}
 \max_n\norm{e_h^n}_0^2
 +\sum_n\tau_n\norm{e_h^n}_U^2
 +\sum_n\norm{e_h^n-e_h^{n-1}}_0^2
 \le C_\nu\left(\norm{e_h^0}_0^2
               +\sum_n\tau_n\norm{\mathcal E_h^n}_{U'}^2\right).
\end{equation}
The initial error is at most $C_gh^2|\omega_0|$, and
Lemma~\ref{lem:rotation-load} makes the remaining term tend to zero.
Thus $W_m\to w$ in $L^2(0,T;U)$ and uniformly in the bulk norm.
Moreover,
\[
 \sum_n\norm{w_h^n-w_h^{n-1}}_0^2
 \le2\sum_n\norm{y_h^n-y_h^{n-1}}_0^2
       +2\sum_n\norm{e_h^n-e_h^{n-1}}_0^2\longrightarrow0,
\]
which gives the centered loss assertion by
\eqref{eq:rotation-loss}.

Finally $U_m=W_m+\omega_{\tau_m}g_{h_m}$, whereas $u=w+\omega g$.
The lifting difference satisfies
\[
 \norm{\omega_{\tau_m}g_{h_m}-\omega g}_{L^2(0,T;U)}
 \le \sqrt T\norm\omega_{L^\infty}\norm{g_{h_m}-g}_U
       +\norm{\omega_{\tau_m}-\omega}_{L^2(0,T)}\norm g_U,
\]
and its supremum bulk norm is bounded by
\[
 \norm\omega_{L^\infty}\norm{g_{h_m}-g}_0
       +\norm{\omega_{\tau_m}-\omega}_{L^\infty(0,T)}\norm g_0.
\]
Equations~\eqref{eq:rotation-lift-error} and
\eqref{eq:omega-time-bounds} show that both expressions tend to zero.
This proves \eqref{eq:rotation-convergence}, including time zero with
the specified initialization.
\end{proof}

\subsection{Quantitative velocity estimate}
Suppose in addition that
$w=u-\omega g\in H^1(0,T;Y)\cap H^1(0,T;V)$ and that
$\lambda\in L^2(0,T;L^2(\Gamma)^2)$ satisfies
\begin{equation}\label{eq:rotation-reaction-identity}
 \ip{\partial_tu}v+a(u,v)+\int_\Gamma\lambda\cdot\gamma_\Gamma v\,ds
        =\dual f v\qquad(v\in U)
\end{equation}
almost everywhere. Define
\begin{align*}
 \mathcal B_m(w,\lambda)={}&
 \sqrt{h_m+\bar\eps_m}
       \bigl(\norm w_{L^\infty(0,T;Y)}
                    +\norm{\partial_tw}_{L^2(0,T;Y)}\bigr)\\
 &+(\sqrt{\bar\eps_m}+\sqrt{\bar\ell_m})
                    \norm\lambda_{L^2(0,T;L^2(\Gamma))}
       +\tau_{\max,m}\norm{\partial_tw}_{L^2(0,T;U)}.
\end{align*}

\begin{theorem}[Energy error with a prescribed rotation lift]
\label{thm:rotation-rate}
Under the preceding regularity and geometric assumptions,
\begin{align}
 &\max_n\norm{u_m^n-u(t_n)}_0
       +\left(\sum_n\tau_m^n\norm{u_m^n-u(t_n)}_U^2\right)^{1/2}
 \nonumber\\
 &\quad\le C\left[
 \mathcal B_m(w,\lambda)
   +h_m^2\{\norm\omega_{L^\infty(0,T)}
                       +\norm{\omega'}_{L^2(0,T)}\}
   +\tau_{\max,m}\norm{\omega'}_{L^2(0,T)}\right].
 \label{eq:rotation-rate}
\end{align}
The constant may depend on the fixed smooth lift, but is independent
of the refinement history and of the number of mesh changes.
\end{theorem}
\begin{proof}
Subtracting the exact lift from
\eqref{eq:rotation-reaction-identity} gives
\[
 \ip{\partial_tw}v+a(w,v)
       +\int_\Gamma\lambda\cdot\gamma_\Gamma v\,ds
       =\dual{\mathcal F}v\qquad(v\in U).
\]
Thus Theorem~\ref{thm:rate} applies to the auxiliary velocities $y_h^n$
used in the preceding proof, with solution $w$, load $\mathcal F$,
and the same $\lambda$. It bounds their nodal and dissipative error
by $C\mathcal B_m(w,\lambda)$. The initial perturbation is at most
$C_gh^2|\omega_0|$. Equations
\eqref{eq:rotation-perturbation-stability} and
\eqref{eq:rotation-load-bound} bound $w_h^n-y_h^n$ by the remaining
terms in \eqref{eq:rotation-rate}. At each node,
\[
 u_h^n-u(t_n)
       =(y_h^n-w(t_n))+(w_h^n-y_h^n)+\omega_n(g_h-g).
\]
The last term has maximum bulk norm at most
$C_gh^2\norm\omega_{L^\infty}$ and weighted $U$ norm at most
$C_gh^2\sqrt T\norm\omega_{L^\infty}$.
The triangle inequality proves the estimate.
\end{proof}

The same right-hand side, with a changed constant, controls
$\norm{U_m-u}_{L^2(0,T;U)}$: the temporal reconstruction error is at
most $\tau_{\max,m}\norm{\partial_tu}_{L^2(0,T;U)}$, and
$\partial_tu=\partial_tw+\omega'g$. As in the homogeneous theorem,
a supremum-in-time error for the piecewise constant reconstruction
additionally incurs an $O(\sqrt{\tau_{\max,m}})$ term under the
stated $H^1$ time regularity; the nodal first-order time estimate in
\eqref{eq:rotation-rate} does not assert a first-order bound for that
reconstruction norm.

\subsection{Rigid interior interpretation and the scope of the energy law}
The affine background-domain problem describes a rigidly rotating
interior at the continuum level if the interior data are chosen
consistently. Specifically suppose
\[
 u_0|_B=\omega(0)r,\qquad
 \dual{f(t)}v=\omega'(t)\ip rv_{L^2(B)}
\]
for every divergence-free $v\in H^1_0(B)^2$, extended by zero to
$\Omega$. A bulk force equal to $\omega'r$ in $B$ is one such choice.
Then $u|_B=\omega r$. To see this, use those interior test functions
in the equation for $w=u-\omega g$. Since $g=r$ in $B$ and $r$ is
affine,
\[
 a_B(r,v)=\nu\int_B\nabla r:\nabla v=0
 \qquad(v\in H^1_0(B)^2).
\]
Thus $w|_B$ has zero trace, zero initial data, and satisfies the
homogeneous interior Stokes evolution. Its energy identity gives
$w|_B=0$. This is a property of the limiting problem; the regularized
discrete solution is not asserted to be exactly rigid inside $B$
at a finite resolution.

Equation~\eqref{eq:rotation-energy} is an exact energy law for the
centered velocity, whereas \eqref{eq:rotation-background-energy}
identifies the work of the prescribed discrete constraint in the
background kinetic-energy balance. The latter is not automatically
an exterior-fluid torque law. The interior kinetic energy and its
matching body-force work must be identified when passing from the
background domain to the physical fluid, and a physical torque
requires the appropriate exterior stress and its approximation.
For example, a rigid rotation has
$a_B(\omega r,\omega r)=2\nu|B|\omega^2$, whereas its symmetric
strain is zero. This local observation does not introduce additional
physical dissipation: for every globally divergence-free
$u\in H^1_0(\Omega)^2$, integration by parts gives
\[
 \nu\int_\Omega|\nabla u|^2
      =2\nu\int_\Omega|D(u)|^2,
 \qquad D(u)=\tfrac12(\nabla u+\nabla u^{\mathsf T}).
\]
If $u=\omega r$ inside $B$, the right-hand side is exactly the
physical exterior strain dissipation. Thus local full-gradient
contributions must not be interpreted separately as physical heating.
A whole-domain reaction such as $\lambda$ in
\eqref{eq:rotation-reaction-identity} includes the interface balance
between the two background subdomains; it is not identified here
with exterior physical traction or a convergent torque. No multiplier
estimate is needed for the centered energy or qualitative convergence.

The result covers prescribed rotation of a disk whose geometric
interface is fixed. It does not cover translation of the disk,
rotation of a nonsymmetric body that changes the interface, an angular
velocity determined by a coupled rigid-body equation, or unrestricted
changes of the lifting function during a run. Those cases would
require additional arguments beyond the two common families
constructed here.