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\begin{document}
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%	
\begin{center}
	{\bf  Response to the reviewer's comments } \\ (\manuscript)
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Dear Professor Tucker,\\

We thank you and the reviewers for the comments and suggestions to improve our manuscript. In the next pages, we replied to each of the comments and included modifications in a new version accordingly. 

We hope that after our response the manuscript can be accepted for publication in \journal. \\

%, in particular for the questions. \\% We will address the comment in the following lines.\\

%In response to the first reviewer, indeed updates on the pressure space definition and notation have been done to correctly defined our problem 
%in the continuum setting. Also comments on the space-time regularity and restrictions for the 
%computational domain were added, e.g. in Remark 6. \\
%
%Our analysis is done throughtout the article using homogeneous Dirichlet boundary conditions and thus our derivations do not contain the mechanical work as expected. 
%We assume such case for matter of convenience since its extension follows by using standard techniques found in literature. Nevertheless, comments 
%about flow intensification were added in Remark 2. \\
%
%Finally, we point out the inaccurate choice of words for our conclusion, which in this new version was updated. \\
%
%In response to the second reviewer, regarding the concern about limitations of first order time discretizations, we studied analogously to the work [1] second order 
%approaches such as Crank-Nicolson and BDF(2) schemes for the monolithic iNSE-ALE problem. To our knowledge, only conditionally energy stable discrete formulations can be 
%obtained with the usual techniques and extensions to pressure correction schemes still lack in unconditional energy stable results. \\
%
%Further, the question of unconditionally energy stable higher order iNSE-ALE formulation still remains open and further research must be done. \\


%We hope that after our response the manuscript can be accepted for publication in \journal. \\

Sincerely yours \\

Reidmen Ar\'{o}stica and Crist\'obal Bertoglio

%\end{document}

%%%% NEW REVIEWER

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\begin{center}
{\bf  Detailed reply to Referee \#1} \\ (\manuscript)
\end{center}

We thank the reviewer for her/his suggestions and comments and we are happy to read the positive comments about our work. Below, we reply in detail to all the raised issues. 
For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{orange}{orange}.


\ReviewComments{p.1 If a function $p$ belongs to the functional space $L^2$, it is not possibble to fix its value at a point. Pressure should be fixed by zero-ing its mean value.}
{
Thanks for pointing this to us,  we corrected it in the new version.
}

\ReviewComments{p.2 On Remark 1: if you address spatial discretizations here, you should mention the problem of convective stabilization. 
For heart chambers, for instance, this is the crucial issue. If you address non-zero boundary conditions and refer to the trace theorem, you should mention
the restrictions of the computational mesh imposed by the trace theorem in finite element spaces.}
{
We thank the reviewer for the comments regarding the restrictions on the computational mesh. Remark 6 has been updated including them.
} 

\ReviewComments{p.2 I do not understand Proposition 1 for a domain with moving boundary: it states that the change of the kinetic energy is spent to viscous dissipation.
However, the motion of the boundary performs a mechanical work, and the energy balance should account flow intensification due to the moving boundary, see, e.g. [12].}
{
We fully agree with the reviewer's comment. In our case, the mechanical work of the boundary does not appear provided the homogeneous Dirichlet boundary conditions originally imposed to the problem. 
In this version, we added a comment regarding such fact in Remark 2.
}

\ReviewComments{p.2 after Remark 3: check carefully classification of all known methods. I suspect that it contains errors. At least, the paper [12] has another set of indices n+1, n, n, n+1 
and this is important since it is proved to be unconditionally stable whereas Corollary 3 states that it has restrictions on the time step.}
{
We thank the reviewer for this comment. We noticed a typo in the discrete formulation (5) which could have induced misundestanding throughout the article. 
We reviewed carefully the discrete formulation in [12], where $k$ there corresponds to $n+1$ in our manuscript. 
Therefore, we updated the indices to (n+1, n, n, n+1) to match the proposed scheme in [12].
}

\ReviewComments{p.4 line -10: moving domain formulation is weird: $X_0 = 0$ is mapped to 1, $X_0$ = 6 is mapped to 1+5.4sin(8$\pi$t), i.e. spans the interval [-4.4, 6.4], which means degeneration
and tangling of the physical domain.}
{
The reviewer is right that $X_0 = 0$ is mapped to 1, nevertheless $X_0 = 6$ is mapped to $6 (1 + 0.9 sin(8 \pi t))$, spanning the interval $[0.6, 11.4]$ thus no mesh degeneration.
}

\ReviewComments{p.4 Fig 1: these curves imply that you lost boundary flow intensification term in the energy balance.}
{
We agree that this would be indeed the general case. However, since homogeneous Dirichlet boundary conditions are used in the numerical experiment, the energy balance does not include intensification terms.
}

\ReviewComments{p.4, first line of Conclusion: the authors are too ambitious to claim that they theoretically analyzed time discretization schemes.
Actually, they suggest hints to a future stability analysis.}
{
We agree that our choice of words was inaccurate. This has been updated.
}

\ReviewComments{General: in mathematical journals, integrals are completed by dx, dt, ds, etc.}
{
Thanks for this remark. It has been updated.
}

\ReviewComments{p1, line 4 of section 2: $J^t$; p2. (4): $w^{**}$ no subscript $_h$ here!; p.4. Boundary conditions are not set in section 5.; p.5 Fig. 1: use $M \star\star$ rather than M**.
p.1. Jacobian, is the standard, p.4. Schwarz, Monolithic.}
{
We acknowledge the comments regarding figure styling and typos. They have been updated in this version.
}
\pagebreak


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\begin{center}
{\bf  Detailed reply to Referee \#2} \\ (\manuscript)
\end{center}

We thank the reviewer for her/his positive impression of the manuscript and for his/her suggestions and comments. Below, we reply in detail to all the raised issues. 
For the sake of convenience, all changes in the revised manuscript version are marked in \textcolor{red}{red}.

%\vspace{-.5cm}

\ReviewComments{My main concern however is the limitation to first order method. In applications I do not think that first order schemes are of much interest. 
At least in FSI, mostly higher BFD schemes [1] (also includin a stability analysis), the fractional step theta scheme [2] or variants of the Crank-Nicolson method [3] 
are applied since they all give second order. Is it possible to include at least one of these schemes in the analysis?}
{
We thank the reviewer for the comment. Analogous to [1] second-order monolithic schemes for the iNSE-ALE problem can be proposed, e.g. using Crank-Nicolson and BDF(2) approaches. However,  as obtained in [1] for the advection-diffusion problem, both CN and BDF(2) are only conditionally  stable, depending on the domain velocity. % and thus out of the scope of this work. 
We believe that obtaining a higher order in time, unconditionally energy stable scheme for the  advection-diffusion equations in ALE form (what includes the iNSE)  still remains an open problem. 
We have now extended Remark 6 in order to explain so and refer to [1] for more details.
}

\ReviewComments{Also, there is much research on pressure correction schemes of higher order, (I guess not for ALE), but it would be worth to extend these 
ideas to ALE formulations.}
{
We agree with the reviewer that there is a lack of higher order fractional-step schemes for the iNSE problem in ALE formalism. In this article, we provide 
a Chorin-Temam approach using our finding for the first order monolithic schemes since unconditional energy stability can be archieved. 

As mentioned in the previous comment, already with second order schemes, e.g. Crank-Nicolson and BDF(2), unconditional energy stability is an open problem for the monolithic case, 
thus its extension to pressure correction schemes can only be expected with conditional energy stability if current techniques are used, a topic beyond the scope of this paper.
}

\ReviewComments{Notation: the index 't' is sometimes a subscript, sometimes a superscript, e.g. in Jt in line 4 of Section 2, but $J^t$ in (1). Maybe this could be unified?}
{
Thanks for pointing that inconsistency out, we acknowledge the lack of unification in the notation. We have updated the notation to consider all time indexes as superscripts.
}

\ReviewComments{Problem description: Do you not require any temporal regularity of the domain velocity w(t)?}
{
We fully agree with the reviewer, a temporal regularity assumption for the domain velocity was lacking. We have updated the regularity of the deformation mapping $\mathcal{X}$ to be at least $\mathcal{C}^1$ is space and time, 
thus for $\frac{\partial \mathcal{X}}{\partial t} := \mathbf{w}$ to be $\mathcal{C}^1$ in space and $\mathcal{C}^0$ in time.
}

\ReviewComments{I think the assumption of an initial pressure p(0)=0 might cause problems. Is this compatible with the configuration that you use in your test case, i.e. 
does it fit to the initial velocity and the right hand side, the Dirichlet data? How do you check that?}
{
Indeed it was a typo. It was removed and updated to the correct 
pressure space of $L^2(\Omega^0)$ functions with zero-mean.
}

\ReviewComments{The notation based on stars is difficult to undestand. I see that it is very general and has advantages. But, if you refer to specific shcemes, I could not undestand 
what exactly is meant without wrtiting it down. Maybe you can give details on the most relevant schemes that you are considering.}
{
We thank the reviewer for the comment and agree with it. Nevertheless it provides advantages when describing a wide range of schemes.
In particular, the items after Remark 4 where we adress specific choices of indices $(\star, \star\star, \ast, \ast\ast)$ are based on relevant schemes applied in different scenarios.
}

\ReviewComments{Proposition 4. What exactly is the role of (7). You derive it by testing with proper functions, thats fine. And I see that many terms cancel for the specific choice of alpha, beta
and star-star. But where does it go to?}
{
Thanks for pointing that out. We notice the lack of a detailed description which has been corrected in this version of the article. (7) was intended to describe the unconditional energy stability obtained
for the monolithic case whenever suitable conditions are applied (in this case: $\alpha = \beta = 1, \star\star = n$). Now, Proposition 4 includes the specific bound obtained from the analysis.
}

\ReviewComments{Numerical examples: Is u(0) a scalar? I guess, 2nd component is zero? Does this initial value fit to a zero pressure at time t=0?}
{
We thank the reviewer for the question. It was a typo in the definition of $u(0)$ that has been updated to the correct expression. 
As guessed by the reviewer, the 2nd component indeed is 0.\\
As clarified above, there is no initial condition for the pressure since it was a typo.
%For the monolithic scheme, only the initial velocity is required for the time iteration whereas for the Chorin-Temam scheme, the initial pressure is computed solving $(FVS)_{0}$. 
%Thus, no direct computation of $p_0$ is required.
}

\ReviewComments{Reference [1] has some html code.}
{
Thanks for pointing that out. It has been corrected.
}

\ReviewComments{Reference [9] has been published in 2018 in Mathematics and Computer in Simulation.}
{
Thanks, it has been updated.
} 

\vspace{1cm}

{
[1] L. Formaggia, F. Nobile. Stability analysis of second-order time
accurate schemes for ALE-FEM, Comp. Meth. Appl. Math. Engrg. 193, 2004

[2] M. Razzaq, H. Damanik, J. Hron, A. Ouazzi, S. Turek, FEM multigrid
techniques for fluid-structure interaction with application to
hemodynamics, Applied Numerical Mathematics, 2012

[3] T. Richter and T. Wick. In T. Carraro and M. Geiger and S. Körkel
and R. Rannacher (Eds.), On time discretizations of fluid-structure
interactions, Multiple Shooting and Time Domain Decomposition
Methods. Springer, 2015
}
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