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@article{Formaggia2004,
doi = {10.1016/j.cma.2003.09.028},
url = {https://doi.org/10.1016/j.cma.2003.09.028},
year = {2004},
month = oct,
publisher = {Elsevier {BV}},
volume = {193},
number = {39-41},
pages = {4097--4116},
author = {Luca Formaggia and Fabio Nobile},
title = {Stability analysis of second-order time accurate schemes for {ALE}{\textendash}{FEM}},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@book{Ern2004,
doi = {10.1007/978-1-4757-4355-5},
url = {https://doi.org/10.1007/978-1-4757-4355-5},
year = {2004},
publisher = {Springer New York},
author = {Alexandre Ern and Jean-Luc Guermond},
title = {Theory and Practice of Finite Elements}
}
@article{FEniCS2015,
author = {Aln\ae{}s, Martin and Blechta, Jan and Hake, Johan and Johansson, August and Kehlet, Benjamin and Logg, Anders and Richardson, Chris and Ring, Johannes and Rognes, Marie E and Wells, Garth N},
language = {eng},
title = {The FEniCS Project Version 1.5},
journal = {Archive of Numerical Software},
volume = {Vol 3},
pages = {Starting Point and Frequency. Year: 2013},
publisher = {University Library Heidelberg},
year = {2015},
copyright = {Authors who publish with this journal agree to the following terms: Authors retain copyright and grant the journal right of first publication with the descriptive part of the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The code part of the work is licensed under a suitable OSI approved Open Source license. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.}
}
@book{Quarteroni2009Cardiovascular,
year = {2009},
publisher = {Springer Milan},
editor = {Luca Formaggia and Alfio Quarteroni and Alessandro Veneziani},
title = {Cardiovascular Mathematics}
}
@book{Richter2017,
year = {2017},
publisher = {Springer International Publishing},
author = {Thomas Richter},
title = {Fluid-structure Interactions}
}
@article{bertoglio2018benchmark,
title={Benchmark problems for numerical treatment of backflow at open boundaries},
author={Bertoglio, Crist{\'o}bal and Caiazzo, Alfonso and Bazilevs, Yuri and Braack, Malte and Esmaily, Mahdi and Gravemeier, Volker and L. Marsden, Alison and Pironneau, Olivier and E. Vignon-Clementel, Irene and A. Wall, Wolfgang},
journal={International journal for numerical methods in biomedical engineering},
volume={34},
number={2},
pages={e2918},
year={2018},
publisher={Wiley Online Library}
}
@article{Bert2018,
author = {Bertoglio, C. and Nu{\~n}ez, R. and Galarce, F. and Nordsletten, D. and Osses, A.},
title = {Relative pressure estimation from velocity measurements in blood flows: State-of-the-art and new approaches},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
volume = {2018;34:e2925. https://doi.org/10.1002/cnm.2925},
year = {2018}
}
@article{Lozovskiy2018,
year = {2018},
month = may,
publisher = {Elsevier {BV}},
volume = {333},
pages = {55--73},
author = {Alexander Lozovskiy and Maxim A. Olshanskii and Yuri V. Vassilevski},
title = {A quasi-Lagrangian finite element method for the Navier{\textendash}Stokes equations in a time-dependent domain},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Liu2018,
year = {2018},
month = aug,
publisher = {Elsevier {BV}},
volume = {337},
pages = {549--597},
author = {Ju Liu and Alison L. Marsden},
title = {A unified continuum and variational multiscale formulation for fluids, solids, and fluid{\textendash}structure interaction},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Nordsletten2008,
year = {2008},
publisher = {Wiley},
volume = {56},
number = {8},
pages = {1457--1463},
author = {D. A. Nordsletten and P. J. Hunter and N. P. Smith},
title = {Conservative and non-conservative arbitrary Lagrangian{\textendash}Eulerian forms for ventricular flows},
journal = {International Journal for Numerical Methods in Fluids}
}
@article{Landajuela2016,
year = {2016},
month = jul,
publisher = {Wiley},
volume = {33},
number = {4},
pages = {e2813},
author = {Mikel Landajuela and Marina Vidrascu and Dominique Chapelle and Miguel A. Fern{\'{a}}ndez},
title = {Coupling schemes for the {FSI} forward prediction challenge: Comparative study and validation},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@article{Burtschell2017,
year = {2017},
month = apr,
publisher = {Elsevier {BV}},
volume = {182},
pages = {313--324},
author = {Bruno Burtschell and Dominique Chapelle and Philippe Moireau},
title = {Effective and energy-preserving time discretization for a general nonlinear poromechanical formulation},
journal = {Computers {\&} Structures}
}
@article{Basting2017,
year = {2017},
month = feb,
publisher = {Elsevier {BV}},
volume = {331},
pages = {312--336},
author = {Steffen Basting and Annalisa Quaini and Sun{\v{c}}ica {\v{C}}ani{\'{c}} and Roland Glowinski},
title = {Extended {ALE} Method for fluid{\textendash}structure interaction problems with large structural displacements},
journal = {Journal of Computational Physics}
}
@article{Deparis2016,
year = {2016},
month = dec,
publisher = {Elsevier {BV}},
volume = {327},
pages = {700--718},
author = {Simone Deparis and Davide Forti and Gwenol Grandperrin and Alfio Quarteroni},
title = {{FaCSI}: A block parallel preconditioner for fluid{\textendash}structure interaction in hemodynamics},
journal = {Journal of Computational Physics}
}
@incollection{Colciago2017,
year = {2017},
month = nov,
publisher = {De Gruyter},
author = {Claudia M. Colciago and Simone Deparis and Davide Forti},
editor = {Stefan Frei and B\"{a}rbel Holm and Thomas Richter and Thomas Wick and Huidong Yang},
title = {7. Fluid-structure interaction for vascular flows: From supercomputers to laptops},
booktitle = {Fluid-Structure Interaction}
}
@article{LeTallec2001,
year = {2001},
month = mar,
publisher = {Elsevier {BV}},
volume = {190},
number = {24-25},
pages = {3039--3067},
author = {P. Le Tallec and J. Mouro},
title = {Fluid structure interaction with large structural displacements},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@phdthesis{smaldone2014,
TITLE = {{Numerical analysis and simulations of coupled problems for the cariovascular system}},
AUTHOR = {Smaldone, Saverio},
SCHOOL = {{L'UNIVERSIT{\'E} PIERRE ET MARIE CURIE - Paris VI }},
YEAR = {2014},
MONTH = Oct,
TYPE = {Theses},
HAL_ID = {tel-01287506},
HAL_VERSION = {v1},
}
@article{Balzani2015,
year = {2015},
month = dec,
publisher = {Wiley},
volume = {32},
number = {10},
pages = {e02756},
author = {Daniel Balzani and Simone Deparis and Simon Fausten and Davide Forti and Alexander Heinlein and Axel Klawonn and Alfio Quarteroni and Oliver Rheinbach and Joerg Schr\"{o}der},
title = {Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@misc{langer2014numerical,
title={Numerical Simulation of Fluid-Structure Interaction Problems with Hyperelastic Models: A Monolithic Approach},
author={Ulrich Langer and Huidong Yang},
year={2014},
eprint={1408.3737},
archivePrefix={arXiv},
primaryClass={math.NA}
}
@article{Langer2018,
doi = {10.1016/j.matcom.2016.07.008},
url = {https://doi.org/10.1016/j.matcom.2016.07.008},
year = {2018},
month = mar,
publisher = {Elsevier {BV}},
volume = {145},
pages = {186--208},
author = {Ulrich Langer and Huidong Yang},
title = {Numerical simulation of fluid{\textendash}structure interaction problems with hyperelastic models: A monolithic approach},
journal = {Mathematics and Computers in Simulation}
}
@misc{failer2020impact,
title={On the Impact of Fluid Structure Interaction in Blood Flow Simulations: Stenotic Coronary Artery Benchmark},
author={Lukas Failer and Piotr Minakowski and Thomas Richter},
year={2020},
eprint={2003.05214},
archivePrefix={arXiv},
primaryClass={physics.comp-ph}
}
@article{Langer2016,
year = {2016},
month = feb,
publisher = {Wiley},
volume = {108},
number = {4},
pages = {303--325},
author = {Ulrich Langer and Huidong Yang},
title = {Robust and efficient monolithic fluid-structure-interaction solvers},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Boffi2004,
year = {2004},
month = oct,
publisher = {Elsevier {BV}},
volume = {193},
number = {42-44},
pages = {4717--4739},
author = {Daniele Boffi and Lucia Gastaldi},
title = {Stability and geometric conservation laws for {ALE} formulations},
journal = {Computer Methods in Applied Mechanics and Engineering}
}
@article{Murea2016,
year = {2016},
month = jun,
publisher = {Wiley},
volume = {109},
number = {8},
pages = {1067--1084},
author = {Cornel Marius Murea and Soyibou Sy},
title = {Updated Lagrangian/Arbitrary Lagrangian-Eulerian framework for interaction between a compressible neo-Hookean structure and an incompressible fluid},
journal = {International Journal for Numerical Methods in Engineering}
}
@article{Hessenthaler2017,
year = {2017},
month = feb,
publisher = {Wiley},
volume = {33},
number = {8},
pages = {e2845},
author = {Andreas Hessenthaler and Oliver R\"{o}hrle and David Nordsletten},
title = {Validation of a non-conforming monolithic fluid-structure interaction method using phase-contrast {MRI}},
journal = {International Journal for Numerical Methods in Biomedical Engineering}
}
@Inbook{Wall2009,
author="Wall, W. A.
and Gerstenberger, A.
and Mayer, U. M.",
editor="Eberhardsteiner, Josef
and Hellmich, Christian
and Mang, Herbert A.
and P{\'e}riaux, Jacques",
title="Advances in Fixed-Grid Fluid Structure Interaction",
bookTitle="ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids",
year="2009",
publisher="Springer Netherlands",
address="Dordrecht",
pages="235--249",
isbn="978-1-4020-9231-2",
doi="10.1007/978-1-4020-9231-2_16",
url="https://doi.org/10.1007/978-1-4020-9231-2_16"
}
@inproceedings{Tallec2003,
author = {Tallec, Le and Hauret, Patrice},
year = {2003},
month = {01},
pages = {},
title = {Energy conservation in fluid-structure interactions}
}
@misc{Wang2020,
title={An energy stable one-field monolithic arbitrary Lagrangian-Eulerian formulation for fluid-structure interaction},
author={Yongxing Wang and Peter K. Jimack and Mark A. Walkley and Olivier Pironneau},
year={2020},
eprint={2003.03819},
archivePrefix={arXiv},
primaryClass={cs.CE}
}

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\documentclass[11pt, a4paper]{article}
%\usepackage[margin=0.6in]{geometry}
\usepackage{fullpage}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage{amsmath,amsfonts}
\usepackage{amsthm,amssymb}
\usepackage{cite}
\usepackage{hyperref}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}[proposition]{Corollary}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
% EDITING COMMANDS
\newcommand{\del}[1]{\sout{#1}}
\renewcommand{\mod}[2]{\sout{#1}$\mapsto$#2}
\newcommand{\RA}[1]{\textcolor{magenta}{RA: #1}}
\newcommand{\CB}[1]{\textcolor{blue}{CB: #1}}
\newcommand{\reva}[1]{\textcolor{orange}{#1}}
\newcommand{\revb}[1]{\textcolor{red}{#1}}
\newcommand{\revc}[1]{\textcolor{cyan}{#1}}
\newcommand{\revs}[1]{\textcolor{olive}{#1}}
\newcommand{\dif}[1]{\,\text{d}\mathbf{#1}}
\title{On monolithic and Chorin-Temam schemes for incompressible flows in moving domains}
\author{
Reidmen Ar\'{o}stica, Crist\'{o}bal Bertoglio \\
Bernoulli Institute, University of Groningen,
9747AG Groningen, The Netherlands
}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
Several time discretization schemes for the incompressible Navier-Stokes equations (iNSE) in moving domains have been proposed.
Here we introduce them in a unified fashion, allowing a common well possedness and time stability analysis. It can be therefore shown that only a particular choice of the numerical scheme ensures such properties. The analysis is performed for monolithic and Chorin-Temam schemes. Results are supported by numerical experiments.
\end{abstract}
\section{Introduction}
%When working with flows from a simulation point of view, several schemes are proposed depending on its applications, suitable for specific requirement e.g. high spatio-temporal resolution, fidelity, stability or fast implementation/simulation times.
%The literature is vast, but to our knowledge there is not a overview trying to summarize their approaches in a single scheme.
Several works have been reported dealing with the numerical solution of the iNSE in moving domains within an Arbitrary Lagrangian Eulerian formulation (ALE), primarily in the context of fluid-solid coupling.
In particular different choices of time discretization have been reported on \cite{Basting2017, Murea2016, Landajuela2016,Lozovskiy2018,smaldone2014,langer2014numerical,LeTallec2001,Liu2018,failer2020impact,Hessenthaler2017}.
% In \cite{Basting2017, Murea2016, Landajuela2016} the FSI within Arbitrary Lagrangian-Eulerian (ALE) formalism in a linearized approach is studied, and \cite{Lozovskiy2018,smaldone2014} also include mass conservation terms. \cite{langer2014numerical,LeTallec2001,Liu2018} propose fully non-linear FSI schemes, and extensions using Crank-Nicholson are used by \cite{failer2020impact,Hessenthaler2017}. \CB{No entiendo la clasificacion de los papers, me parece que le falta un poco de coherencia. No deberian tambien haber mas papers, o estos son todos? }
%Thougthful
To the best of the authors knowledge, only a few monolithic schemes have been thoroughly analyzed, e.g. in \cite{Lozovskiy2018, Burtschell2017, LeTallec2001, smaldone2014}, while no analysis has been reported for Chorin-Temam (CT) methods.
The goal of this work is therefore to assess well-posedness and unconditional energy balance of the iNSE-ALE for all reported monolithic and CT discretization schemes within a single formulation.
% Maybe we need to add the works of \cite{Boffi2004} for general surveys of ALE schemes.
%State-of-the art: monolithic (what is proven), CT (not proven, just "used") \\
%, which as it will be seen later on, holds under some restrictions with help of consistent stabilization terms.
%The findings for the monolithic case are then used to introduce a Chorin-Temam scheme, for which unconditional energy stability holds.
The reminder of this paper is structured as follows:
Section \ref{sec:continuous_problem} provides the continuous problem that will be studied. Section \ref{sec:monolithic_schemes} introduces a general monolithic scheme that characterizes several approaches used in literature, well-posedness and energy stability are studied and discussed. Section \ref{sec:chorin_temam_schemes} introduces the Chorin-Temam schemes where time stability is analyzed. Finally, Section \ref{sec:numerical_examples} provides numerical examples testing our results.
\section{The continuous problem}
\label{sec:continuous_problem}
In the following, let us consider a domain $\Omega^0 \subset \mathbb{R}^d$ with $d = 2,3$ and a deformation mapping $\mathcal{X}: \mathbb{R}^d\times \mathbb{R}_{+}\mapsto\mathbb{R}^d$ that defines the time evolving domain $\Omega^t := \mathcal{X}(\Omega^0, t)$.
We assume $\mathcal{X}$ a $\mathcal{C}^1$ mapping in both coordinates, 1-to-1 with $\mathcal{C}^1$ inverse. We denote $\mathbf{X} \in \mathbb{R}^d$ the cartesian coordinate system in $\Omega^0$ and $\mathbf{x}^t := \mathcal{X}(\mathbf{X}, t)$ the one in $\Omega^t$, by $F^t := \frac{\partial \mathbf{x}^t}{\partial \mathbf{X}}$ the deformation gradient, $H^t := (F^t)^{-1}$ its inverse and $J^t := det(F^t)$ its Jacobian.
Similarly, $Grad(\mathbf{\mathbf{f}}) := \frac{\partial \mathbf{f}}{\partial \mathbf{X}}$, $Div(\mathbf{f}) := \frac{\partial}{\partial \mathbf{X}} \cdot \mathbf{f}$ denote the gradient and divergence operators respectively and $\epsilon^t(\mathbf{f}) := \frac{1}{2}( Grad(\mathbf{f}) H^t + (H^t)^{T} Grad(\mathbf{f})^{T})$ the symmetric gradient, for $\mathbf{f}$ a well-defined vector function.
By $\mathbf{H}^1_0 (\Omega^0)$ we denote the standard Sobolev space of vector fields $\mathbf{u}$ defined in $\Omega^0$ with values in $\mathbb{R}^d$ such that $\mathbf{u} = \mathbf{0}$ on $\partial \Omega^0$, by $L^2_0(\Omega^0)$ the standard square integrable space of functions $r$ defined in $\Omega^0$ with values in $\mathbb{R}$ s.t. $\int_{\Omega^0} r \, \text{d}\mathbf{X} = 0$ and $T > 0$ a final time.
We consider the weak form of the iNSE in ALE form \cite[Ch. 5]{Richter2017}: Find $(\mathbf{u}(t), p(t)) \in \mathbf{H}^{1}_{0}(\Omega^0) \times L^2_0(\Omega^0)$ for $t \in (0, T)$ with $\mathbf{u}(0) = \mathbf{u}_{init}$ s.t.
\begin{equation}
\label{eq:continuous_formulation}
\begin{aligned}
\int_{\Omega^0} \rho J^t \frac{\partial \mathbf{u} }{\partial t} \cdot \mathbf{v} + \rho J^t Grad(\mathbf{u}) H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{v} + J^t 2\mu \, \epsilon^t(\mathbf{u}):\epsilon^t(\mathbf{v}) \dif{X} & \\
- \int_{\Omega^0} Div(J^t H^t \mathbf{v}) p \dif{X} + \int_{\Omega^0} Div(J^t H^t \mathbf{u})q \dif{X} &= 0%& \\
\end{aligned}
\end{equation}
for all $(\mathbf{v}, q) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$, $\mathbf{u}_{init} \in \mathbf{H}^1_0(\Omega^0)$ given initial and $\mathbf{w} := \frac{\partial \mathcal{X}}{\partial t}$ time-varying domain velocities. For the sake of simplicity, we omit the time-dependency on the fields $\mathbf{u}, p$.
Notice that the velocity flow at time $t$ is given by $\mathbf{u} \circ \mathcal{X}^{-1}(\cdot, t)$.
\begin{proposition}
\label{prop:energy_continuous} \cite[Chap. 9]{Quarteroni2009Cardiovascular}
Provided $(\mathbf{u}(t), p(t)) \in \mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ a solution of Problem \eqref{eq:continuous_formulation}, % and $\mathbf{u}_{\mathbf{g}}(t) \in \mathbf{H}^1_{\mathbf{g}}(\Omega^0)$ the extension of $g \in L^2(\partial \Omega^0)$
the following energy balance holds:
\begin{equation}
\label{eq:continuous_energy_estimate}
\begin{aligned}
\frac{\partial}{\partial t} \int_{\Omega^0} \frac{\rho}{2} J^t \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X} =& - \int_{\Omega^0} J^t 2\mu \vert \epsilon^t (\mathbf{u}) \vert^2 \, \text{d}\mathbf{X}. %- \int_{\partial\Omega_0} \rho J_t F^{-1}_{t} (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S} + \mathcal{D}(\mathbf{u}; \mathbf{u}_{\mathbf{g}})
\end{aligned}
\end{equation}
\end{proposition}
\begin{remark}
Proposition \ref{prop:energy_continuous} makes use of the \textit{Geometric Conservation Law} (GCL) $\frac{\partial J^t}{\partial t} = Div\left( J^t F_{t}^{-1} \mathbf{w}\right)$.
\end{remark}
\begin{remark}
In the general case with non-homogeneous Dirichlet boundary conditions,
the energy balance also includes flow intensification due to the moving boundary. In such case, the intensification term appearing on the energy balance \eqref{eq:continuous_energy_estimate} in given by:
\begin{equation}
\label{eq:extension_energy_continuous}
\int_{\partial\Omega^0} \rho \frac{\vert \mathbf{u} \vert^2}{2} J^t H^t (\mathbf{u} - \mathbf{w}) \cdot \mathbf{N} \, \text{d}\mathbf{S}
\end{equation}
where $\mathbf{N} \in \mathbb{R}^d$ denotes the outward normal.
\end{remark}
\begin{remark}
Although Dirichlet boundary conditions are used throughout this work, it can be extended straightforwardly to the Neumann case by including the so called \textit{backflow stabilizations}, see e.g. \cite{bertoglio2018benchmark}.
\end{remark}
\section{Monolithic schemes (first order in time)}
\label{sec:monolithic_schemes}
Most of the numerical schemes for Problem \eqref{eq:continuous_formulation} reported in the literature are first order and can be written as follows.
Let $(t^n)_{n \in \mathbb{N}}$ be a uniform discretization of the time interval $(0, T)$ with step size $\tau > 0$
and let $H^n := H^{t^n}, J^n := J^{t^n}, w^n := w(t^n)$ be discrete sequences.
Given a conforming finite element space $\mathbf{V} \times Q$ of $\mathbf{H}^1_0(\Omega^0) \times L^2_0(\Omega^0)$ for velocity and pressure fields, the discrete problem of interest reads:
Find $(\mathbf{u}^{n+1}, p^{n+1}) \in \mathbf{V} \times Q$ s.t.
\begin{equation}
\label{eq:discretized_monolithic_formulation}
\mathcal{A}(\mathbf{u}^{n+1},\mathbf{v}) - \mathcal{B}(\mathbf{v},p^{n+1}) + \mathcal{B}(\mathbf{u}^{n+1},q) = \mathcal{F}(\mathbf{v}) \quad \forall (\mathbf{v}, q) \in \mathbf{V} \times Q
\end{equation}
being
\begin{equation}
\label{eq:lhs_bilinear_form_A}
\begin{aligned}
\mathcal{A} (\mathbf{u}, \mathbf{v}) := & \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} \rho J^{\star} Grad(\mathbf{u}) H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} + \int_{\Omega^0} J^{\star} 2\mu \epsilon^{\star}(\mathbf{u}):\epsilon^{\star}(\mathbf{v}) \, \text{d}\mathbf{X} \\
& + \alpha \int_{\Omega^0} \frac{\rho}{2} \left( \frac{J^{n+1} - J^{n}}{\tau} - Div\left( J^{\star} H^{\star} \mathbf{w}^{\ast\ast} \right) \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X} + \beta \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} \mathbf{u}^{\ast} \right) \mathbf{u} \cdot \mathbf{v} \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
with $\alpha, \beta \in \{0, 1\}$ given parameters, and
%The matrix $B \in \mathbb{R}^{m^{Q}\times m^{\mathbf{V}}}$ and vector $F_n \in \mathbb{R}^n$ correspond also to the discretization of the bilinear and linear forms:
\begin{equation}
\label{eq:remaining_forms}
\begin{aligned}
\mathcal{B}(\mathbf{u}, q) & := \int_{\Omega^0} Div\left( J^{\star} H^{\star} \mathbf{u} \right) q \, \text{d}\mathbf{X} \quad \forall q \in Q, \quad \mathcal{F}(\mathbf{v}) := \int_{\Omega^0} \rho \frac{J^{\star\star}}{\tau} \mathbf{u}^n \cdot \mathbf{v} \, \text{d}\mathbf{X} \quad \forall \mathbf{v} \in \mathbf{V}
\end{aligned}
\end{equation}
\begin{remark}
The term multiplying $\alpha$ is the discrete residual of GCL, while the one multiplying $\beta$ is a strongly consistent term vanishing for incompressible velocity fields.
\end{remark}
Formulation \eqref{eq:discretized_monolithic_formulation} contains a wide family of reported methods:
\begin{itemize}
\item Using $\alpha = \beta = 0$: $(\star, \star\star, \ast, \ast\ast) = (n, n, n+1, n)$ is used in \cite{Basting2017}, $(\star, \star\star, \ast, \ast \ast) = (n, n, n, n)$ in \cite{Murea2016} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n+1, n+1)$ in \cite{Langer2016}, and $(\star, \star\star, \ast, \ast\ast) = (n+1, n+1, n, n+1)$ in \cite{Landajuela2016}.
\item Using $\alpha = \beta = 1$: $(\star, \star\star, \ast, \ast \ast) = (n+1, n, n, n+1)$ in \cite{Lozovskiy2018}, $(\star, \star\star,\ast, \ast\ast) = (n+1, n, n, n)$ in \cite{smaldone2014} and $(\star, \star\star, \ast, \ast\ast) = (n+1, n, n+1, n+1)$ in \cite{LeTallec2001, Wang2020}.
\end{itemize}
\begin{proposition}
\label{prop:monolithic_schemes}
By assuming well-posed, orientation-preserving deformation mappings, i.e. $(J^n)_{n \in \mathbb{N}}$ bounded in $L^{\infty}(\Omega^0)$, $J^n > 0$ for each $n \geq 0$, Problem \eqref{eq:discretized_monolithic_formulation} has unique solution for inf-sup stable finite element spaces if $\left( 2J^{\star\star} + J^{n+1} - J^{n} \right) > 0$ and $\alpha = \beta = 1$.
\end{proposition}
\begin{proof}
%The matrix system \eqref{eq:matrix_formulation} is solvable whenever $A_n^{-1}$ exists and $BA^{-1}_nB^{T}$ is invertible.
Since all operators are bounded and inf-sup stable elements are used for velocity and pressure, it is enough to ensure that the bilinear form $\mathcal{A}$ is coercive.
%Thus its enough to ensure $\text{rank}(B) = m$ and $A_n$ positive definite.
%As standard in literature, let us evaluate \eqref{eq:lhs_bilinear_form_A} in $\mathbf{v} = \mathbf{u}$ and $q = p$. Integrating by parts the convective term and joining expressions, the following equality holds:
Indeed:
\begin{equation}
\label{eq:proof_monolithic_case}
\begin{aligned}
\mathcal{A}(\mathbf{u}, \mathbf{u}) = & \int_{\Omega^0} \frac{J^{\star}}{2\tau} \left( \frac{2 J^{\star\star}}{J_{\star}} + \alpha \frac{J^{n+1} - J^{n}}{J^{\star}} \right) \vert \mathbf{u} \vert^2 + % \int_{\Omega^0}
J^{\star} 2\mu \vert \epsilon^{\star}(\mathbf{u}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\bigg( J^{\star} H^{\star} \big( (\beta -1) \mathbf{u}^{\ast} - (\alpha-1) \mathbf{w}^{\ast\ast} \big) \bigg) \vert \mathbf{u} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
being the last quantity strictly positive under the stated assumptions.
\end{proof}
\begin{remark}
The extension of Proposition \ref{prop:monolithic_schemes} to the case with non-homogeneous Dirichlet boundary conditions follows from the trace theorem by assuming $\Omega^0$ a Lipschitz bounded open set \cite{Ern2004}.
\end{remark}
\begin{corollary}
Assuming $\alpha = \beta = 1$, Problem \eqref{eq:discretized_monolithic_formulation} is well posed when:
\begin{itemize}
\item $3J^{n+1} - J^{n} > 0$ if $\star \star = n+1$, i.e. a restriction on the time step size.
\item $J^{n+1} + J^{n} > 0$ if $\star \star = n$, i.e. no restriction on the time step size, since we assume orientation preserving deformation mappings.
\end{itemize}
No restrictions apply to $\star,\ast, \ast\ast$.
\end{corollary}
\begin{proposition}
\label{prop:energy_estimate_monolithic}
Under assumptions of Proposition \ref{prop:monolithic_schemes} and $\alpha = \beta = 1, \star\star = n$, the scheme \eqref{eq:discretized_monolithic_formulation} is unconditionally energy stable with energy estimate:
\begin{equation}
\label{eq:energy_estimate_monolithic}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \rho \frac{J^n}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X}
= -\int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star}(\mathbf{u}^{n+1}) \vert^2 \, \text{d} \mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{n} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d} \mathbf{X}.
\end{equation}
\end{proposition}
\begin{proof}
By setting $\mathbf{v} = \mathbf{u}^{n+1}$ in the bi-linear form \eqref{eq:lhs_bilinear_form_A}, $q = p^{n+1}$ in forms \eqref{eq:remaining_forms} and manipulating terms as standard in literature, the energy equality follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} +
\int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^n) \vert \mathbf{u}^n \vert^2 \, \text{d}\mathbf{X} & \\
& - \int_{\Omega^0} 2\mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div(J^{\star} H^{\star} (\mathbf{u}^{\ast} - \mathbf{w}^{\ast\ast})) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2} \alpha \frac{J^{n+1} - J^{n}}{\tau} \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star} (\beta \mathbf{u}^{\ast} - \alpha \mathbf{w}^{\ast\ast}) \right) \vert \mathbf{u}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Thus, for $\alpha=\beta=1$ and $\star\star = n$ the result follows.
\end{proof}
\begin{remark}
This works focuses on first-order schemes in time.
The reason is that second order schemes, although stable in fixed domain, has been shown to be only conditionally stable in ALE form, as it was shown in \cite{Formaggia2004} for the advection-diffusion problem for Crank-Nicolson (CN) and BDF(2).
Therefore, we do not analyze here the schemes used in \cite{ failer2020impact, Hessenthaler2017,Tallec2003} -- based on CN and used in the context of fluid-solid interaction -- since their analysis repeats from \cite{Formaggia2004}.
Also in the same context, some authors have used the generalized $\alpha$-methods since it is a popular scheme for elastodynamics \cite{Liu2018}. However, there is no reported stability analysis even for the the fixed domain setting,
and its stability properties are usually assumed to be transferred from the linear setting.
\end{remark}
\section{Chorin-Temam schemes}
\label{sec:chorin_temam_schemes}
In the following, we describe a family of Chorin-Temam (CT) schemes for the iNSE-ALE problem, as we did for the monolithic case. %Such description keeps the freedom of choice for certain coefficients, which must be restricted to ensure unconditional stability.
Given $\mathbf{\widetilde V} $ a conforming space of $\mathbf{H}^1_0 (\Omega^0)$ and $ \widetilde Q$ a conforming space of $ L^2_0 (\Omega^0) \cap H^1(\Omega^0)$, $\tilde{\mathbf{u}}^{0} \in \mathbf{\widetilde V}$, for $n\geq0$:% the proposed two-step CT schemes reads
\begin{enumerate}
\item \textbf{Pressure-Projection Step $(\text{PPS})_{n}$}
%Seeks a pressure term allowing the projection of $\tilde{\mathbf{u}}^{n+1}$ in the space of divergence-free solutions, in the form:
Find $p^{n} \in \widetilde{Q}$ s.t.
\begin{equation}
\label{eq:chorin_temam_pps}
\int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} Grad(p^{n}) H^{\circ} : Grad(q) H^{\circ} \, \text{d}\mathbf{X} = - \int_{\Omega^0} Div\left( J^{\circ} H^{\circ} \tilde{\mathbf{u}}^{n}\right) q \, \text{d}\mathbf{X} \quad \forall q \in \widetilde{Q}
\end{equation}
\item \textbf{Fluid-Viscous Step $(\text{FVS})_{n+1}$} %Seeks a tentative velocity field $\tilde{\mathbf{u}}^{n+1}$ with pressure given explicitly:
Find $\tilde{\mathbf{u}}^{n+1} \in \mathbf{\widetilde{V}}$ s.t.
\begin{equation}
\label{eq:chorin_temam_fvs}
\begin{aligned}
\int_{\Omega^0} \rho J^{\star\star} \frac{\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^n}{\tau} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \rho J^{\star} Grad(\tilde{\mathbf{u}}^{n+1}) H^{\star} (\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast}) \cdot \mathbf{v} \, \text{d}\mathbf{X} & \\
+ \int_{\Omega^0} J^{\star} 2 \mu \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) : \epsilon^{\star} (\mathbf{v}) \, \text{d}\mathbf{X}
- \int_{\Omega^0} Div(J^{\circ \circ} H^{\circ \circ} \mathbf{v}) p^n \, \text{d}\mathbf{X} & \\
+ \int_{\Omega^0} \frac{\rho}{2} \frac{J^{n+1} - J^{n}}{\tau} \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2} Div\left( J^{\star} H^{\star}(\tilde{\mathbf{u}}^{n} - \mathbf{w}^{\ast\ast})\right) \tilde{\mathbf{u}}^{n+1} \cdot \mathbf{v} \, \text{d}\mathbf{X} & = 0 \quad \forall \mathbf{v} \in \mathbf{\widetilde{V}}
\end{aligned}
\end{equation}
% In particular, the corrected velocity is obtained through the update:
% \begin{equation}
% \begin{aligned}
% \mathbf{u}^{n+1} & = \tilde{\mathbf{u}}^{n+1} - \frac{\tau}{\rho} Grad(p^{n+1}) F^{-1}_{\circ + 1} \text{ in } \Omega_0
% \end{aligned}
% \end{equation}
\end{enumerate}
The following energy estimate can be obtained under suitable conditions:
\begin{proposition}
\label{prop:energy_estimate_chorin_temam}
Under assumptions $\circ = \circ\circ = \star \star = n$, the solution to scheme \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is unconditionally stable, i.e.
%is unconditionally energy stable if $\circ = \circ \circ = \star\star = n$, without condition over $\star$. Moreover, the energy estimate is given by
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d} \mathbf{X} \leq & - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{n} \frac{\tau}{2\rho} \vert Grad(p^n) H^{n} \vert^2 \, \text{d}\mathbf{X} .
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
As standard in literature, let us take $\mathbf{v} = \tilde{\mathbf{u}}^{n+1}$ in $(\text{FVS})_{n+1}$, and $q = p^n$ in $(\text{PPS})_{n}$. Adding both equalities and rewriting expressions, it follows:
\begin{equation}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} = & \int_{\Omega^0} \frac{\rho}{2\tau}(J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau}(J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^n \right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
Bounding the first divergence term using integration by parts and Cauchy-Schwarz inequality, it follows
\begin{equation}
\begin{aligned}
\int_{\Omega^0} Div\left( J^{\circ\circ} H^{\circ\circ} (\tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n}) \right) p^n \, \text{d}\mathbf{X} & \leq \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{x}
\end{aligned}
\end{equation}
Thus, the following energy estimate can be obtained:
\begin{equation}
\label{eq:energy_estimate_proof}
\begin{aligned}
\int_{\Omega^0} \rho \frac{J^{n+1}}{2\tau} \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} \rho \frac{J^{n}}{2\tau} \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \leq & \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{\star\star}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
+ \int_{\Omega^0} \frac{\rho}{2\tau} (J^{\star\star} - J^{n}) \vert \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} J^{\star\star} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} - \int_{\Omega^0} J^{\star} 2\mu \vert \epsilon^{\star} (\tilde{\mathbf{u}}^{n+1}) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\rho}{2\tau} J^{\circ\circ} \vert \tilde{\mathbf{u}}^{n+1} - \tilde{\mathbf{u}}^{n} \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} \frac{\tau}{2\rho} J^{\circ\circ} \vert (H^{\circ\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& + \int_{\Omega^0} Div\left( (J^{\circ\circ} H^{\circ\circ} - J^{\circ} H^{\circ}) \tilde{\mathbf{u}}^{n}\right) p^n \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\tau}{\rho} J^{\circ} \vert (H^{\circ})^{T} Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
& - \int_{\Omega^0} \frac{\rho}{2\tau} (J^{n+1} - J^{n}) \vert \tilde{\mathbf{u}}^{n+1} \vert^2 \, \text{d}\mathbf{X}
\end{aligned}
\end{equation}
From estimate \eqref{eq:energy_estimate_proof} it follows that whenever $\circ = \circ \circ = \star \star = n$ unconditional energy stability is attained, where $\star$ remains free of choice.
\end{proof}
\section{Numerical examples}
\label{sec:numerical_examples}
We consider a rectangular domain with opposite vertices $\{ (0, -1), (6, 1) \}$ where the iNSE-ALE formulation \eqref{eq:continuous_formulation} will be simulated over the interval $(0,\, 2) \, [s]$ with non-zero initial condition of the form $ \mathbf{u}(0) := \big(\gamma (1 - \mathbf{X}_1^2) \mathbf{X}_0 (6 - \mathbf{X}_0), 0\big) ,\, \gamma = 0.001$.
The domain is deformed using $\mathcal{X}(\mathbf{X}, t) := \big( (1 + 0.9 sin(8 \pi t)) \mathbf{X}_0,\, \mathbf{X}_1 \big)$. %, i.e. an oscillation with initial expansion and frequency of $4 \pi$.
Discretization setup for Formulation \eqref{eq:discretized_monolithic_formulation} and \eqref{eq:chorin_temam_pps}-\eqref{eq:chorin_temam_fvs} is done choosing a time step $\tau = 0.01$ and space triangulation with elements diameter $h \approx 0.01 $, implemented through FEniCS \cite{FEniCS2015} using Python for interface and postprocessing.
To exemplify the theoretical results from previous sections, four schemes are taken into account. Monolithic (M) Formulation \eqref{eq:discretized_monolithic_formulation} is taken with linearized convective term and implicit treatment, i.e., $(\star, \ast, \ast\ast) = (n+1, n, n+1)$ where for $\star\star$ we consider two choices, denoted $\text{M}\, \star\star = n$ and $\text{M}\, \star\star = n+1$.
For both cases the space discretization is carried out with $ \mathbf{V}/Q = [\mathbb{P}_2]^d/\mathbb{P}_1$ Lagrange finite elements. Similarly, Chorin-Temam (CT) scheme \eqref{eq:chorin_temam_fvs}-\eqref{eq:chorin_temam_pps} is taken with linearized convective term and implicit treatment, i.e. $(\star, \ast\ast, \circ, \circ\circ) = (n+1, n+1, n, n)$ with $\star\star$ as before, denoting each scheme by $\text{CT}\, \star\star = n$ and $\text{CT}\, \star\star = n+1$ with space discretization done through $\mathbf{\widetilde{V}}/\widetilde{Q} = [\mathbb{P}_1]^d/\mathbb{P}_1$ elements.
In all cases homogeneous (equal to $\mathbf{0}$) boundary conditions are imposed for the velocity, zero-mean on the pressure and $\alpha=\beta=1$.
The results are assessed using time-dependent normalized parameters $ \hat{\delta}_{\text{M}}:= \delta_{\text{M}}/E_{st}^{\star}, \hat{\delta}_{\text{CT}}:= \delta_{\text{CT}}/E_{st}^{\star}$ defined as: %, where $E_{st}^{\star}$ denote the \CB{\mod{strain energy}{viscous dissipation}} and $\delta_{M}, \delta_{CT}$ residual errors defined as:
\begin{equation}
\label{eq:energy_error}
\begin{aligned}
\delta_{M}^{n+1} &:= D^{n+1} + E_{st}^{\star} + \int_{\Omega_0} \frac{\rho J^{\star\star}}{2\tau} \vert \mathbf{u}^{n+1} - \mathbf{u}^{n} \vert^2 \, \text{d}\mathbf{X} , \quad \delta_{CT}^{n+1} := D^{n+1} + E^{\star}_{st} + \int_{\Omega_0} \frac{\tau J^{\circ}}{2 \rho} \vert (H^{\circ})^T Grad(p^n) \vert^2 \, \text{d}\mathbf{X} \\
D^{n+1} &:= \int_{\Omega_0} \frac{\rho}{2\tau} \left( J^{n+1} \vert \mathbf{u}^{n+1} \vert^2 - J^n \vert \mathbf{u}^n \vert^2 \right) \, \text{d}\mathbf{X} , \quad E^{\star}_{st} = \int_{\Omega_0} 2 \mu J^{\star} \vert \epsilon^{\star} (\mathbf{u}^{n+1}) \vert^2 \, \text{d}\mathbf{X}.
\end{aligned}
\end{equation}
Figure \ref{fig:delta_hat_oscs} shows $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ values for each tested scheme. Propositions \ref{prop:energy_estimate_monolithic} and \ref{prop:energy_estimate_chorin_temam} are confirmed since $\hat{\delta}_{\text{M}}=0$ and $\hat{\delta}_{\text{CT}} \leq 0$ if $\star\star = n$. For $\star \star = n+1$, peaks appearing throughout the simulation are defined by the sign change of domain velocity, i.e. in the change from expansion to contraction.
Importantly, the spurious numerical energy rate related to discretization of the GCL condition appear to be positive in expansion, therefore being a potential source of numerical instabilities.
%Moreover the expected energy decay from Propositions \ref{prop:energy_estimate_monolithic}, \ref{prop:energy_estimate_chorin_temam} for the choice $\star \star = n$ in $\text{M, CT}$ schemes is obtained.
%\CB{Recall that $\hat{\delta}_{\text{M}}, \hat{\delta}_{\text{CT}}$ in \eqref{eq:energy_error} are taken in such form since on scheme $\text{M}\, \star\star = n$ it follows $\hat{\delta}_{MT} = 0$ and similarly on $\text{CT}\, \star\star = n$ it follows $\hat{\delta}_{\text{CT}} \sim 0$, defining the base case.}
\begin{figure}[!hbtp]
\centering
\includegraphics[width=\textwidth]{figs/Comparison_Delta_Hat_Value_GCL_True_solver_LU.png}
\caption{Summary of the numerical experiment in terms of energy balance. Left: Monolithic residual error values $\hat{\delta}_{\text{M}}$;
Right: Chorin-Temam residual error values $\hat{\delta}_{\text{CT}}$. %Both cases are simulated with deformation mapping $\mathcal{A}$ defined previously and shown on the interval $[0, 1] \, [s]$.}
}
\label{fig:delta_hat_oscs}
\end{figure}
\section{Conclusion}
Several reported time discretization schemes for the iNSE-ALE have been reviewed and analyzed in terms of their well posedness at each time step and time stability. The stability analysis is confirmed by numerical experiments. For the monolithic case, two schemes lead to well-posed energy-stable problems whenever $\alpha=\beta=1$ with $\star \star = n$ as studied in \cite{LeTallec2001, Lozovskiy2018, smaldone2014, Wang2020}.
To the best of the authors knowledge, the unconditionally stable Chorin-Temam scheme derived in this work has not been reported yet. %, and moreover the numerical experiments studied here validate the energy stable propositions.
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