\section{Cartesian spaces and a common constrained approximation family}
\label{sec:spaces}

\subsection{A concrete two-level conforming construction}
Let $h=1/M$, where $M$ is a positive integer, and let $\mathcal T_h$ be
the uniform partition of $\Omega$ into squares of side $h$. On a rectangle,
write $\mathbb Q_3$ for polynomials of degree at most three in each variable.
Let $S_h\subset H^2_0(\Omega)$ be the clamped Cartesian bicubic Hermite
space. Its local degrees of freedom at each vertex are the value,
the two first derivatives, and the mixed derivative $\partial_{xy}$.
Matching these degrees of freedom gives a globally $C^1$ function. The
boundary degrees of freedom are set to zero, so that both the function
and its normal derivative vanish on $\partial\Omega$. This is the
standard Cartesian Bogner--Fox--Schmit construction; see, for example,
\citet{bfs2020} for its tensor-product description and interpolation
estimates.

For a coarse cell $K\in\mathcal T_h$, divide $K$ into four squares of side
$h/2$. Define $B_h(K)$ to consist of the $C^1$ piecewise bicubics on this
subdivision whose value and gradient vanish on $\partial K$. Extending
such a function by zero gives an element of $H^2_0(\Omega)$. Equivalently,
all Hermite degrees of freedom at the eight boundary vertices are zero
and the four degrees of freedom at the center vertex are free. Hence
$\dim B_h(K)=4$. For any set $\mathcal A\subset\mathcal T_h$, put
\begin{equation}\label{eq:cartesian-spaces}
 S_{h,\mathcal A}=S_h+\sum_{K\in\mathcal A}B_h(K),\qquad
 X_{h,\mathcal A}=\curl S_{h,\mathcal A},\qquad
 \curl\psi=(\partial_y\psi,-\partial_x\psi).
\end{equation}
Every $X_{h,\mathcal A}$ is a finite-dimensional subspace of $U$.
Incompressibility holds exactly, since $\divg\curl\psi=0$ in
distributions for $\psi\in H^2$. The zero boundary gradient of the
streamfunction gives the outer homogeneous velocity condition.

This construction retains the full coarse space and adds local fine-grid
degrees of freedom. All the spaces are contained in the uniform $h/2$
master space, but only cells in $\mathcal A$ carry the additional center
degrees of freedom. Adding or deleting an arbitrary cell from
$\mathcal A$ is a genuine local refinement or coarsening of this
specified conforming space. The zero value and gradient of each local
enrichment on the coarse-cell boundary prevent a conformity defect at
a coarse--fine interface. We do not identify this space with the full
space of every possible hanging-node implementation.

For a regularization width $\eps$ and a panel space $\Lambda_\mathcal P$,
define
\begin{equation}\label{eq:spatial-Z}
 Z_{h,\mathcal A,\eps,\mathcal P}
 =\{v_h\in X_{h,\mathcal A}:b_\eps(v_h,\mu)=0
       \text{ for every }\mu\in\Lambda_\mathcal P\}.
\end{equation}
These are exact surface-moment constraints. Linear independence of their
rows is neither assumed nor needed for the velocity approximation
arguments below.

\subsection{A family present in every configuration}
Let $\bar\eps$ be an upper bound for every regularization width in one
run. Define the union of coarse cells
\begin{equation}\label{eq:collar-cells}
 \begin{aligned}
 D_{h,\bar\eps}
 &=\bigcup\{K\in\mathcal T_h:
       K\cap\{x:\dist(x,\Gamma)\le\bar\eps\}\ne\varnothing\},\\
 K_{h,\bar\eps}
 &=\{v_h\in\curl S_h:v_h=0\text{ on }D_{h,\bar\eps}\}.
 \end{aligned}
\end{equation}
The equalities defining $K_{h,\bar\eps}$ are polynomial identities on
specified coarse cells, and therefore define a finite-dimensional
linear space. They impose zero \emph{velocity}; they do not impose zero
streamfunction on the immersed collar. The latter would unnecessarily
exclude circulation modes.

If $0<\eps\le\bar\eps$, every kernel support centered on $\Gamma$ lies
inside $D_{h,\bar\eps}$. Thus
\begin{equation}\label{eq:common-inclusion}
 K_{h,\bar\eps}\subset V,
 \qquad
 K_{h,\bar\eps}\subset
 Z_{h,\mathcal A,\eps,\mathcal P}
 \quad\text{for every }\mathcal A,\mathcal P,
 \quad 0<\eps\le\bar\eps.
\end{equation}
In particular, an orthogonal transfer onto any of these constraint
spaces fixes every element of $K_{h,\bar\eps}$. The next result shows
that this common family is large enough to approximate the physical
solution space.

\begin{lemma}[Common recovery family]\label{lem:recovery}
Let $h_m\to0$ and $\bar\eps_m\to0$. For each $w\in V$ there exist
$w_m\in K_{h_m,\bar\eps_m}$ such that
\begin{equation}\label{eq:recovery}
 \norm{w_m-w}_U\longrightarrow0.
\end{equation}
The sequence $w_m$ is independent of the refinement sets, the panel
partitions, and the individual regularization widths used during run
$m$, provided those widths do not exceed $\bar\eps_m$. No relation
between $h_m$ and $\bar\eps_m$ is required.
\end{lemma}
\begin{proof}
The restrictions of $w$ to $B$ and $F$ belong to the zero-trace
solenoidal $H^1$ spaces on these domains. The standard solenoidal density
theorem on bounded Lipschitz domains, applied separately in the two
domains, gives smooth divergence-free fields compactly supported in
$B$ and $F$ that converge to these restrictions in $H^1$; see
\citet[Section III.4]{galdi2011}. Their zero extensions and sums form
a sequence $w^{(j)}\in C_c^\infty(\Omega)^2$ converging to $w$ in $U$.
For each $j$, $w^{(j)}$ vanishes on a neighborhood of $\Gamma$ and is
globally divergence free. This domainwise construction is valid in the
multiply connected exterior region; it does not assert that the
streamfunction has the same constant on its different boundary
components.

For a fixed such field $w^{(j)}$, define explicitly
\[
 \psi^{(j)}(x,y)=\int_0^y w^{(j)}_1(x,s)\,ds.
\]
Since $\divg w^{(j)}=0$ and $w^{(j)}_2(x,0)=0$, one has
$\partial_y\psi^{(j)}=w^{(j)}_1$ and
$\partial_x\psi^{(j)}=-w^{(j)}_2$. The zero outer velocity implies
$\psi^{(j)}=0$ and $\nabla\psi^{(j)}=0$ on the outer square. On the
connected collar of $\Gamma$, $\psi^{(j)}$ is a constant, which need
not be zero. Thus $\psi^{(j)}$ is a smooth clamped streamfunction.

Let $I_h\psi^{(j)}\in S_h$ be its bicubic Hermite interpolant.
The Cartesian Hermite interpolation estimate, for example
\citet[equation (52), with $m=2$]{bfs2020}, gives
\begin{equation}\label{eq:BFS-interpolation}
 \norm{I_h\psi^{(j)}-\psi^{(j)}}_{H^2(\Omega)}
 \le C h^2\norm{\psi^{(j)}}_{H^4(\Omega)}.
\end{equation}
If $w^{(j)}$ vanishes whenever $\dist(x,\Gamma)<r_j$, and
$\bar\eps_m+\sqrt2h_m<r_j$, every coarse cell in
$D_{h_m,\bar\eps_m}$ lies in this constant-streamfunction collar.
Hermite interpolation reproduces the constant exactly on such a cell.
Consequently,
$\curl I_{h_m}\psi^{(j)}\in K_{h_m,\bar\eps_m}$ for all sufficiently
large $m$, and it converges to $w^{(j)}$ in $U$.

Choose $j=j(m)$ tending to infinity sufficiently slowly that both this
collar inclusion and the interpolation convergence hold, while
$\norm{w^{(j(m))}-w}_U\to0$. The triangle inequality proves
\eqref{eq:recovery}.
\end{proof}

The collar is used to construct comparison functions. The actual
discrete solution is constrained only by \eqref{eq:spatial-Z}; it is
not required to vanish on $D_{h,\bar\eps}$.

\subsection{A quantitative collar construction}
For a velocity $w$ that is smooth separately in the body and the fluid,
write
\[
 \mathcal M(w)=\norm w_{W^{3,\infty}(B)}
             +\norm w_{W^{3,\infty}(F)}.
\]
All restrictions and traces in the next lemma are understood in the
Sobolev sense. A globally $H^1$ velocity with a jump in its first
derivatives across $\Gamma$ is allowed.

\begin{lemma}[Quantitative common approximation]\label{lem:collar}
There are constants $c_0,C>0$, depending only on the fixed geometry,
such that the following holds. If $h+\bar\eps<c_0$, there is a linear
operator $Q_{h,\bar\eps}$ from
$V\cap[W^{3,\infty}(B)^2\times W^{3,\infty}(F)^2]$
into $K_{h,\bar\eps}$ satisfying
\begin{equation}\label{eq:collar-rate}
 \norm{Q_{h,\bar\eps}w-w}_{H^1(\Omega)}
 \le C\sqrt{h+\bar\eps}\,\mathcal M(w).
\end{equation}
For fixed $h,\bar\eps$, this operator commutes with the weak time
derivative whenever $w$ belongs to the Bochner $H^1$ space in the
stated piecewise regularity class, with
$w(t),\partial_t w(t)\in V$ almost everywhere. Hence the same estimate
applies almost everywhere to the time derivative of the approximation
error, with $\mathcal M(\partial_t w)$ on the right.
\end{lemma}
\begin{proof}
The signed-distance tubular coordinates used below are standard for a smooth
fixed interface; see also the geometric setup in \citet{bfs2020}.
The flattening and width-dependent estimates are derived here.
As in the preceding proof, set
$\psi(x,y)=\int_0^y w_1(x,s)\,ds$. Then
$\curl\psi=w$, $\psi\in H^2_0(\Omega)$, and $\psi$ is separately
$W^{4,\infty}$ on $B$ and $F$. Its piecewise $W^{4,\infty}$ norm is
bounded by $C\mathcal M(w)$. Since $\nabla\psi=0$ on the connected
interface, $\psi|_\Gamma=c$ for one scalar $c$. Taylor's formula along
the interface normals gives, on either side and in a fixed tubular
neighborhood,
\begin{equation}\label{eq:quadratic-vanishing}
 |\psi-c|\le C d^2\mathcal M(w),\qquad
 |\nabla\psi|\le C|d|\mathcal M(w),
\end{equation}
where $d$ is the signed distance to $\Gamma$.

Choose a fixed smooth cutoff $\chi$ with $\chi(s)=1$ for $|s|\le1$
and $\chi(s)=0$ for $|s|\ge2$. Put
\begin{equation}\label{eq:delta-choice}
 \delta=4(\bar\eps+\sqrt2h),\qquad
 \psi_\delta=\psi-\chi(d/\delta)(\psi-c)
\end{equation}
inside the tubular neighborhood, and set $\psi_\delta=\psi$ elsewhere.
Choose $c_0$ sufficiently small that the $2\delta$ collar is contained
in that neighborhood and disjoint from $\partial\Omega$. This preserves
the clamped outer condition. The modified streamfunction equals $c$
on $|d|\le\delta$, and thus removes the original interface derivative
jumps completely: $\psi_\delta\in H^4(\Omega)\cap H^2_0(\Omega)$.

The difference $\psi_\delta-\psi$ is supported in a set of area
$O(\delta)$. By \eqref{eq:quadratic-vanishing}, the zeroth, first, and
second derivatives of this difference are bounded pointwise by
$C\delta^2\mathcal M(w)$, $C\delta\mathcal M(w)$, and
$C\mathcal M(w)$, respectively. Consequently,
\begin{equation}\label{eq:flatten-H2}
 \norm{\psi_\delta-\psi}_{H^2(\Omega)}
 \le C\sqrt\delta\,\mathcal M(w).
\end{equation}
On the transition region $\delta<|d|<2\delta$, the fourth derivatives
of the cutoff product are bounded by
$C\delta^{-2}\mathcal M(w)$. Indeed, the terms with four and three
cutoff derivatives are multiplied by $\psi-c=O(\delta^2)$ and
$\nabla\psi=O(\delta)$, respectively; the terms with two cutoff
derivatives multiply bounded second derivatives of $\psi$. The
remaining terms are no larger for sufficiently small $\delta$.
Derivatives of the signed-distance function are bounded on the fixed
tubular neighborhood. Since this transition region also has area
$O(\delta)$, we obtain
\begin{equation}\label{eq:flatten-H4}
 \norm{\psi_\delta}_{H^4(\Omega)}
 \le C(1+\delta^{-3/2})\mathcal M(w).
\end{equation}

Define $Q_{h,\bar\eps}w=\curl I_h\psi_\delta$. A coarse cell meeting
the $\bar\eps$ collar is contained in the collar of width
$\bar\eps+\sqrt2h<\delta$, where $\psi_\delta=c$. Exact constant
reproduction therefore proves $Q_{h,\bar\eps}w\in K_{h,\bar\eps}$.
Combining \eqref{eq:BFS-interpolation}, \eqref{eq:flatten-H2}, and
\eqref{eq:flatten-H4} gives
\[
 \norm{Q_{h,\bar\eps}w-w}_{H^1(\Omega)}
 \le C\bigl(\sqrt\delta+h^2(1+\delta^{-3/2})\bigr)\mathcal M(w).
\]
Because $\delta\ge4\sqrt2h$ and
$\delta\le C(h+\bar\eps)$, this is \eqref{eq:collar-rate}.

Finally, the construction $w\mapsto\psi$, extraction of its constant
trace $c$, multiplication by the fixed cutoff, and Hermite
interpolation are all linear. For fixed $h,\bar\eps$, the resulting
operator is bounded from the stated piecewise regularity class,
equipped with the norm $\mathcal M$, into $H^1(\Omega)^2$; this follows
also from \eqref{eq:collar-rate} and the continuous inclusion of that
class in $H^1(\Omega)^2$. The geometry, $\delta$, and the coarse grid
are fixed throughout the run. A bounded linear map commutes with
Bochner weak differentiation, which proves the last assertion.
\end{proof}

The error in \eqref{eq:collar-rate} is expressed in the retained coarse
mesh size. It proves a robust approximation property, rather than
optimality of an adaptive marking strategy. It remains valid when the
regularization support intersects coarse--fine boundaries because the
comparison function is already zero on the entire support. No
uniform estimate for the boundary multiplier is inferred from this
construction.