\section{Convergence with arbitrarily frequent mesh changes}\label{sec:convergence}
We index complete computations by $m$. The coarse mesh size is $h_m$;
the time partition, the selected refined cells, the regularization widths,
and the surface partitions may all depend on $m$. Write
\[
 \bar\eps_m=\max_{0\le n\le N_m}\eps_m^n,\qquad
 \bar\ell_m=\max_{0\le n\le N_m}\ell_m^n,\qquad
 \tau_{\max,m}=\max_{1\le n\le N_m}\tau_m^n.
\]
The maxima include the choices used to define the initial projector.
Let $U_m(t)=u_m^n$ on $(t_{n-1},t_n]$ and $U_m(0)=u_m^0$.
The following theorem is qualitative; it requires no regularity of an
interface multiplier.

\begin{theorem}[Strong velocity convergence]\label{thm:convergence}
Let $u_0\in H$ and $f\in L^2(0,T;U')$. Suppose
\begin{equation}\label{eq:parameters-to-zero}
 h_m+\bar\eps_m+\bar\ell_m+\tau_{\max,m}\longrightarrow0.
\end{equation}
At every time level choose any admissible refinement patch and impose
the exact moment constraints in \eqref{eq:Z}. Let $u_m^0=P_m^0u_0$ and
let $u_m^n$ solve \eqref{eq:scheme}, with the time averages of $f$.
Then, for the solution $u$ of \eqref{eq:weak},
\begin{equation}\label{eq:strong-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}
In addition, the total numerical loss tends to zero:
\begin{equation}\label{eq:loss-to-zero}
 \sum_{n=1}^{N_m}\left\{
   \norm{u_m^n-P_m^nu_m^{n-1}}_0^2+
   \norm{(I-P_m^n)u_m^{n-1}}_0^2\right\}
 \longrightarrow0.
\end{equation}
There is no restriction relating $N_m$ or the number of patch changes
to $h_m$, $\bar\eps_m$, or $\bar\ell_m$, beyond
\eqref{eq:parameters-to-zero}.
\end{theorem}

\begin{proof}
\emph{Initial approximation and compactness.}
The common recovery property of Lemma~\ref{lem:recovery} implies that,
for every $w\in V$, there are $w_m$ belonging to every $Z_m^n$ with
$w_m\to w$ in $U$. Since $V$ is dense in $H$ by definition, the
best-approximation property of $P_m^0$ gives
\begin{equation}\label{eq:initial-convergence}
 \norm{u_m^0-u_0}_0\longrightarrow0.
\end{equation}
Indeed, first approximate $u_0$ in $L^2$ by a fixed element of $V$,
and then use its common recovery sequence as a competitor for $P_m^0$.

The energy bound \eqref{eq:energybound} gives, along a subsequence,
\[
 U_m\rightharpoonup \widetilde u\quad\hbox{in }L^2(0,T;U),
 \qquad
 U_m\overset{*}{\rightharpoonup}\widetilde u
       \quad\hbox{in }L^\infty(0,T;L^2(\Omega)^2).
\]
By Corollary~\ref{cor:slip}, $\gamma_\Gamma U_m\to0$ strongly in
$L^2(0,T;L^2(\Gamma))$. The bounded trace operator therefore gives
$\widetilde u\in L^2(0,T;V)$.

\emph{The evolution equation from common test functions.}
Fix $w\in V$ and a common recovery sequence $w_m\to w$ in $U$.
Define the continuous, piecewise affine scalar function $g_m$ by its
nodal values $g_m(t_n)=\ip{u_m^n}{w_m}$. Lemma~\ref{lem:cancel} gives
\begin{equation}\label{eq:scalar-evolution}
 g_m'(t)=\dual{f_m(t)}{w_m}-a(U_m(t),w_m)
       \quad\hbox{for a.e. }t,
\end{equation}
where $f_m(t)=f_m^n$ on the $n$th time interval.
The derivatives $g_m'$ are uniformly bounded in $L^2(0,T)$, while
$g_m(0)\to\ip{u_0}{w}$. Thus $g_m$ is relatively compact in
$C([0,T])$. Also,
\begin{equation}\label{eq:scalar-reconstruction}
 \sup_{0\le t\le T}
 \left|\ip{U_m(t)}w-g_m(t)\right|
 \le C\norm{w-w_m}_0+
       \sqrt{\tau_{\max,m}}\norm{g_m'}_{L^2(0,T)}
 \longrightarrow0.
\end{equation}
Time averaging on partitions with maximum step tending to zero gives
$f_m\to f$ strongly in $L^2(0,T;U')$.
Passing to the limit in \eqref{eq:scalar-evolution}, first on a
countable dense family in $V$ and then by continuity, yields
\[
 \frac{d}{dt}\ip{\widetilde u(t)}w
   =\dual{f(t)}w-a(\widetilde u(t),w),\qquad
 \ip{\widetilde u(0)}w=\ip{u_0}w
       \quad(w\in V).
\]
Consequently $\partial_t\widetilde u\in L^2(0,T;V')$ and
$\widetilde u$ is the solution of \eqref{eq:weak}. Uniqueness identifies
$\widetilde u=u$. It also implies, for every fixed $w\in V$,
\begin{equation}\label{eq:uniform-moments}
 \sup_{0\le t\le T}
 \left|\ip{U_m(t)-u(t)}w\right|\longrightarrow0.
\end{equation}
The same assertion holds for every $w\in H$, by density and the uniform
bulk norm bound. The weak convergence and moment conclusions hold for
the whole sequence, since every subsequence has the same possible limit.

\emph{Strong convergence in the dissipative norm.}
The terminal moment convergence implies
\begin{equation}\label{eq:terminal-lower}
 \norm{u(T)}_0^2\le\liminf_m\norm{u_m^{N_m}}_0^2.
\end{equation}
This assertion does not require $u_m^{N_m}$ to lie in $H$: test against
elements of the dense subset $V\subset H$ and take the supremum over
unit bulk norm tests. Summing \eqref{eq:energy} gives
\begin{equation}\label{eq:terminal-discrete-energy}
\tfrac12\norm{u_m^{N_m}}_0^2
 +\int_0^T a(U_m,U_m)\,dt+\tfrac12 D_m
 =\tfrac12\norm{u_m^0}_0^2+\int_0^T\dual{f_m}{U_m}\,dt,
\end{equation}
where $D_m=\sum_{n=1}^{N_m}\norm{u_m^n-u_m^{n-1}}_0^2$.
The right-hand side tends to
$\tfrac12\norm{u_0}_0^2+\int_0^T\dual f u\,dt$:
use \eqref{eq:initial-convergence}, strong convergence of $f_m$, and
weak convergence of $U_m$ in $L^2(0,T;U)$. By
\eqref{eq:continuous-energy}, this limit is
$\tfrac12\norm{u(T)}_0^2+\int_0^T a(u,u)\,dt$.
Weak lower semicontinuity for the dissipation, \eqref{eq:terminal-lower},
and $D_m\ge0$ therefore force
\begin{equation}\label{eq:energy-component-convergence}
 \int_0^T a(U_m,U_m)\,dt\longrightarrow\int_0^T a(u,u)\,dt,
 \qquad
 \norm{u_m^{N_m}}_0\longrightarrow\norm{u(T)}_0,
 \qquad D_m\longrightarrow0.
\end{equation}
Since $a(v,v)=\nu\norm v_U^2$, weak convergence together with norm
convergence proves $U_m\to u$ in $L^2(0,T;U)$ by the Hilbert-space
criterion in \citet[Proposition 3.32]{brezis2011}.
Equation~\eqref{eq:lossdecomposition} then proves
\eqref{eq:loss-to-zero}.

There is no hidden terminal component orthogonal to $H$.
To see this explicitly, let $Q_H$ be the bulk orthogonal projection onto
$H$. Equation~\eqref{eq:uniform-moments} gives
$Q_Hu_m^{N_m}\rightharpoonup u(T)$ in $H$. Hence
\[
 \norm{u(T)}_0\le\liminf_m\norm{Q_Hu_m^{N_m}}_0
 \le\limsup_m\norm{Q_Hu_m^{N_m}}_0
 \le\lim_m\norm{u_m^{N_m}}_0=\norm{u(T)}_0.
\]
It follows that $Q_Hu_m^{N_m}\to u(T)$ strongly and
$\norm{(I-Q_H)u_m^{N_m}}_0\to0$.

\emph{Uniform convergence in the bulk norm.}
Strong convergence in $L^2(0,T;U)$ implies
\[
 a(U_m,U_m)\longrightarrow a(u,u)\quad\hbox{in }L^1(0,T),
 \qquad
 \dual{f_m}{U_m}\longrightarrow\dual f u
                  \quad\hbox{in }L^1(0,T).
\]
The corresponding cumulative integrals converge uniformly in their
upper limits. The discrete energy identity at any node, after discarding
the nonnegative accumulated loss, and the continuous energy identity
therefore give
\[
 \limsup_m\max_n
 \left(\norm{u_m^n}_0^2-\norm{u(t_n)}_0^2\right)\le0.
\]
By continuity of $u:[0,T]\to H$ and
$\tau_{\max,m}\to0$, this upper bound also holds with arbitrary
$t$ and $U_m(t)$. The image $u([0,T])$ is a compact subset of $H$.
Using a finite bulk-norm net for this image in
\eqref{eq:uniform-moments} proves
\[
 \sup_{0\le t\le T}
 \left|\ip{U_m(t)-u(t)}{u(t)}\right|\longrightarrow0.
\]
Finally, expand
\[
 \norm{U_m(t)-u(t)}_0^2
 =\norm{U_m(t)}_0^2-\norm{u(t)}_0^2
  -2\ip{U_m(t)-u(t)}{u(t)}.
\]
The preceding two bounds prove uniform bulk convergence.
\end{proof}

\begin{remark}[Why no remeshing count occurs]
The cancellation uses a fixed family of comparison functions common to
all spaces of one computation, not a bound for each mesh-transfer defect.
This differs from the projection-difference terms in
\citet[Theorems 3.1 and 3.4]{brenner2014} and the remeshing contributions
in \citet{gawlik2015}.
The dense family is supplied by the retained coarse grid and the collar
construction. A collection of unrelated shifted grids need not have such
a family. Thus Theorem~\ref{thm:convergence} does not assert convergence
for arbitrary stable interpolations or for arbitrary changing spaces.
\end{remark}